2.2 Linear expressions

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{{Vald flik|[[2.2 Linjära uttryck|Theory]]}}
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{{Selected tab|[[2.2 Linear expressions|Theory]]}}
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{{Ej vald flik|[[2.2 Exercises|Exercises]]}}
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{{Not selected tab|[[2.2 Exercises|Exercises]]}}
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'''Learning outcomes:'''
'''Learning outcomes:'''
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After this section, you will have learned how to:
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After this section you will have learned how to:
*Solve algebraic equations, which after simplification results in first degree equations.
*Solve algebraic equations, which after simplification results in first degree equations.
*Convert between the forms ''y'' = ''kx'' + ''m'' and ''ax'' + ''by'' + ''c'' = 0.
*Convert between the forms ''y'' = ''kx'' + ''m'' and ''ax'' + ''by'' + ''c'' = 0.
*Sketch straight lines from their equation.
*Sketch straight lines from their equation.
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* Solve geometric problems which contain straight lines.
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* Solve geometric problems that contain straight lines.
*Sketch regions defined by linear inequalities and determine the area of these regions.
*Sketch regions defined by linear inequalities and determine the area of these regions.
}}
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== First degree equations ==
== First degree equations ==
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To solve first degree equations (also known as linear equations) we perform calculation on both sides simultaneously, which gradually simplifies the equation and ultimately leads to <math>x</math> being alone on one side of the equation.
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To solve first degree equations (also known as linear equations) we perform calculations on both sides simultaneously. This gradually simplifies the equation and ultimately leads to <math>x</math> being alone on one side of the equation.
<div class="exempel">
<div class="exempel">
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Subtract <math>3</math> from both sides
Subtract <math>3</math> from both sides
:<math>x+3-3=7-3</math>.
:<math>x+3-3=7-3</math>.
-
The left-hand side then simplifies to <math>x</math> , and we get
+
The left-hand side then simplifies to <math>x</math> and we get
:<math>x=7-3=4</math>.</li>
:<math>x=7-3=4</math>.</li>
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Divide both sides by <math>3</math>
Divide both sides by <math>3</math>
:<math>\frac{3x}{3} = \frac{6}{3}\,</math>.
:<math>\frac{3x}{3} = \frac{6}{3}\,</math>.
-
After having cancelled <math>3</math> on the left-hand side, we have
+
After having cancelled <math>3</math> on the left-hand side we have
:<math> x=\frac{6}{3} = 2</math>.</li>
:<math> x=\frac{6}{3} = 2</math>.</li>
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First we subtract <math>1</math> from both sides to get <math>2x</math> on its own on the left-hand side
First we subtract <math>1</math> from both sides to get <math>2x</math> on its own on the left-hand side
:<math>2x=5-1</math>.<br/>
:<math>2x=5-1</math>.<br/>
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Then we divide both sides by <math>2</math> and get the answer
+
We then divide both sides by <math>2</math> to get the answer
:<math>x = \frac{4}{2} = 2</math>.</li>
:<math>x = \frac{4}{2} = 2</math>.</li>
</ol>
</ol>
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A first degree equation can be written in the normal form <math>ax=b</math>. The solution is then simply <math>x=b/a</math> (we must assume that <math>a\not=0</math>).
A first degree equation can be written in the normal form <math>ax=b</math>. The solution is then simply <math>x=b/a</math> (we must assume that <math>a\not=0</math>).
-
The possible difficulties that may occur when you solve a first degree equation are thus not in themselves the form of the solution but rather the simplifications that may be needed to achieve the normal form. Below are a couple examples that have in common that an equation can be simplified to linear normal form and thus has a unique solution.
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The possible difficulties that may occur when you solve a first degree equation are thus not in themselves the form of the solution, but rather the simplifications that may be needed to achieve the normal form. Below are a couple examples in which the equation can be simplified to a linear normal form, thus having a unique solution.
<div class="exempel">
<div class="exempel">
'''Example 2'''
'''Example 2'''
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Since <math>x</math> occurs on both the left- and right-hand sides we subtract <math>2x</math> from both sides
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Since <math>x</math> occurs on both the left and right hand sides we subtract <math>2x</math> from both sides
-
{{Fristående formel||<math>2x-3-2x=5x+7-2x</math>}}
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{{Displayed math||<math>2x-3-2x=5x+7-2x</math>}}
and now <math>x</math> only appears on the right-hand side
and now <math>x</math> only appears on the right-hand side
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{{Fristående formel||<math>-3 = 3x+7 \; \mbox{.}</math>}}
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{{Displayed math||<math>-3 = 3x+7 \; \mbox{.}</math>}}
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Now we subtract 7 from both sides
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We now subtract 7 from both sides
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{{Fristående formel||<math>-3 -7 = 3x +7-7</math>}}
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{{Displayed math||<math>-3 -7 = 3x +7-7</math>}}
and get <math>3x</math> on its own on the right-hand side
and get <math>3x</math> on its own on the right-hand side
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{{Fristående formel||<math>-10=3x\,\mbox{.}</math>}}
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{{Displayed math||<math>-10=3x\,\mbox{.}</math>}}
The last step is to divide both sides by <math>3</math>
The last step is to divide both sides by <math>3</math>
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{{Fristående formel||<math>\frac{-10}{3} = \frac{3x}{3}</math>}}
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{{Displayed math||<math>\frac{-10}{3} = \frac{3x}{3}</math>}}
-
and this gives that
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giving
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{{Fristående formel||<math>x=-\frac{10}{3}\,\mbox{.}</math>}}
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{{Displayed math||<math>x=-\frac{10}{3}\,\mbox{.}</math>}}
</div>
</div>
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'''Example 3'''
'''Example 3'''
-
Solve for <math>x</math> in the equation <math>ax+7=3x-b</math>.
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Solve for <math>x</math> the equation <math>ax+7=3x-b</math>.
By subtracting <math>3x</math> from both sides
By subtracting <math>3x</math> from both sides
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{{Fristående formel||<math>ax+7-3x=3x-b-3x</math>}}
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{{Displayed math||<math>ax+7-3x=3x-b-3x</math>}}
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{{Fristående formel||<math>ax+7-3x=\phantom{3x}{}-b\phantom{{}-3x}</math>}}
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{{Displayed math||<math>ax+7-3x=\phantom{3x}{}-b\phantom{{}-3x}</math>}}
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and then subtract <math>7</math>
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and then subtracting <math>7</math>
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{{Fristående formel||<math>ax+7-3x -7=-b-7</math>}}
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{{Displayed math||<math>ax+7-3x -7=-b-7</math>}}
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{{Fristående formel||<math>ax\phantom{{}+7}{}-3x\phantom{{}-7}{}=-b-7</math>}}
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{{Displayed math||<math>ax\phantom{{}+7}{}-3x\phantom{{}-7}{}=-b-7</math>}}
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We have gathered together all the terms that contain <math>x</math> on the left-hand side and all other terms on the right-hand side. Since the terms on the left-hand side have <math>x</math> as a common factor <math>x</math> can be factored out
+
we have gathered together all the terms that contain <math>x</math> on the left-hand side and all other terms on the right-hand side. Since the terms on the left-hand side have <math>x</math> as a common factor <math>x</math>, they can be factored out
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{{Fristående formel||<math>(a-3)x = -b-7\; \mbox{.}</math>}}
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{{Displayed math||<math>(a-3)x = -b-7\; \mbox{.}</math>}}
Divide both sides with <math>a-3</math> giving
Divide both sides with <math>a-3</math> giving
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{{Fristående formel||<math>x= \frac{-b-7}{a-3}\; \mbox{.}</math>}}
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{{Displayed math||<math>x= \frac{-b-7}{a-3}\; \mbox{.}</math>}}
</div>
</div>
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It is not always obvious that you are dealing with a first degree equation. In the following two examples simplifications turn the original equation into a first degree equation.
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It is not always obvious that we are dealing with a first degree equation. In the following two examples simplifications turn the original equation into a first degree equation.
<div class="exempel">
<div class="exempel">
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Expand the quadratic expressions on both sides
Expand the quadratic expressions on both sides
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{{Fristående formel||<math>x^2-6x+9+3x^2=4x^2+28x+49\,\mbox{,}</math>}}
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{{Displayed math||<math>x^2-6x+9+3x^2=4x^2+28x+49\,\mbox{,}</math>}}
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{{Fristående formel||<math>4x^2-6x+9=4x^2+28x+49\,\mbox{.}</math>}}
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{{Displayed math||<math>4x^2-6x+9=4x^2+28x+49\,\mbox{.}</math>}}
Subtract <math>4x^2</math> from both sides
Subtract <math>4x^2</math> from both sides
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{{Fristående formel||<math>-6x +9 = 28x +49\; \mbox{.}</math>}}
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{{Displayed math||<math>-6x +9 = 28x +49\; \mbox{.}</math>}}
Add <math>6x</math> to both sides
Add <math>6x</math> to both sides
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{{Fristående formel||<math>9 = 34x +49\; \mbox{.}</math>}}
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{{Displayed math||<math>9 = 34x +49\; \mbox{.}</math>}}
Subtract <math>49</math> from both sides
Subtract <math>49</math> from both sides
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{{Fristående formel||<math>-40=34x\; \mbox{.}</math>}}
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{{Displayed math||<math>-40=34x\; \mbox{.}</math>}}
Divide both sides by <math>34</math>
Divide both sides by <math>34</math>
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{{Fristående formel||<math>x = \frac{-40}{34}= - \frac{20}{17}\; \mbox{.}</math>}}
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{{Displayed math||<math>x = \frac{-40}{34}= - \frac{20}{17}\; \mbox{.}</math>}}
</div>
</div>
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Collect both terms to one side
Collect both terms to one side
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{{Fristående formel||<math>\frac{x+2}{x^2+x}-\frac{3}{2+3x}= 0\; \mbox{.}</math>}}
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{{Displayed math||<math>\frac{x+2}{x^2+x}-\frac{3}{2+3x}= 0\; \mbox{.}</math>}}
Convert the terms so that they have the same denominator
Convert the terms so that they have the same denominator
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{{Fristående formel||<math>\frac{(x+2)(2+3x)}{(x^2+x)(2+3x)}-\frac{3(x^2+x)}{(2+3x)(x^2+x)}= 0</math>}}
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{{Displayed math||<math>\frac{(x+2)(2+3x)}{(x^2+x)(2+3x)}-\frac{3(x^2+x)}{(2+3x)(x^2+x)}= 0</math>}}
and simplify the numerator
and simplify the numerator
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{{Fristående formel||<math>\frac{(x+2)(2+3x)-3(x^2+x)}{(x^2+x)(2+3x)} = 0,</math>}}
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{{Displayed math||<math>\frac{(x+2)(2+3x)-3(x^2+x)}{(x^2+x)(2+3x)} = 0,</math>}}
-
{{Fristående formel||<math>\frac{3x^2+8x+4-(3x^2+3x)}{(x^2+x)(2+3x)} = 0,</math>}}
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{{Displayed math||<math>\frac{3x^2+8x+4-(3x^2+3x)}{(x^2+x)(2+3x)} = 0,</math>}}
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{{Fristående formel||<math>\frac{5x +4}{(x^2+x)(2+3x)} = 0\,\mbox{.}</math>}}
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{{Displayed math||<math>\frac{5x +4}{(x^2+x)(2+3x)} = 0\,\mbox{.}</math>}}
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This equation only is satisfied when the numerator is equal to zero (whilst the denominator is not equal to zero);
+
This equation is only satisfied when the numerator is equal to zero (whilst the denominator is not equal to zero);
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{{Fristående formel||<math>5x+4=0</math>}}
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{{Displayed math||<math>5x+4=0</math>}}
which gives that <math>\,x = -\frac{4}{5}</math>.
which gives that <math>\,x = -\frac{4}{5}</math>.
</div>
</div>
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Functions such as
Functions such as
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{{Fristående formel||<math>y = 2x+1</math>}}
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{{Displayed math||<math>y = 2x+1</math>}}
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{{Fristående formel||<math>y = -x+3</math>}}
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{{Displayed math||<math>y = -x+3</math>}}
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{{Fristående formel||<math>y = \frac{1}{2} x -5 </math>}}
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{{Displayed math||<math>y = \frac{1}{2} x -5 </math>}}
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are examples of linear functions, and they generally can be put into the form
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are examples of linear functions. They have the general form
<div class="regel">
<div class="regel">
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{{Fristående formel||<math>y = kx+m</math>}}
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{{Displayed math||<math>y = kx+m</math>}}
</div>
</div>
where <math>k</math> and <math>m</math> are constants.
where <math>k</math> and <math>m</math> are constants.
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The graph of a linear function is always a straight line and the constant <math>k</math> indicates the slope of the line with respect to the <math>x</math>-axis and <math>m</math> gives the <math>y</math>-coordinate of the point where the line intersects the <math>y</math>-axis.
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The graph of a linear function is always a straight line. The constant <math>k</math> indicates the slope of the line with respect to the <math>x</math>-axis and <math>m</math> gives the <math>y</math>-coordinate of the point where the line intersects the <math>y</math>-axis.
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<center>{{:2.2 - Figur - Linjen y = kx + m}}</center>
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<center>{{:2.2 - Figure - The line y = kx + m}}</center>
<center><small>The line ''y'' = ''kx'' + ''m'' has slope ''k'' and cuts the ''y''-axis at (0,''m'')</small></center>
<center><small>The line ''y'' = ''kx'' + ''m'' has slope ''k'' and cuts the ''y''-axis at (0,''m'')</small></center>
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The constant <math>k</math> is called the slope and means that a unit change in the positive <math>x</math>-direction along the line gives <math>k</math> units change in the positive <math>y</math>-direction. Thus if
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The constant <math>k</math> is called the slope and states that given a unit change in the positive <math>x</math>-direction, along the line there is a <math>k</math> unit change in the positive <math>y</math>-direction. Thus if
*<math>k>0\,</math> the line slopes upwards,
*<math>k>0\,</math> the line slopes upwards,
*<math>k<0\,</math> the line slopes downwards.
*<math>k<0\,</math> the line slopes downwards.
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For a horizontal line (parallel to the <math>x</math>-axis) <math>k=0</math> whereas a vertical line (parallel to the <math>y</math>-axis) does not have a <math>k</math> value ( a vertical line cannot be written in the form <math>y=kx+m</math>).
+
For a horizontal line (parallel to the <math>x</math>-axis) <math>k=0</math>, whereas a vertical line (parallel to the <math>y</math>-axis) does not have a <math>k</math> value ( a vertical line cannot be written in the form <math>y=kx+m</math>).
<div class="exempel">
<div class="exempel">
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<ol type="a">
<ol type="a">
<li> Sketch the line <math>y=2x-1</math>. <br/><br/>
<li> Sketch the line <math>y=2x-1</math>. <br/><br/>
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Comparing with the standard equation <math>y=kx+m</math> we see that <math>k=2</math> and <math>m=-1</math>. This means that the line's slope is <math>2</math> and that it cuts the <math>y</math>-axis at <math>(0,-1)</math>. See the figure below to the left.</li>
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Comparing with the standard equation <math>y=kx+m</math>, we see that <math>k=2</math> and <math>m=-1</math>. This gives the line's slope as <math>2</math> and that it cuts the <math>y</math>-axis at <math>(0,-1)</math>. See the figure below-left.</li>
<li>Sketch the line <math>y=2-\tfrac{1}{2}x</math>.<br/><br/>
<li>Sketch the line <math>y=2-\tfrac{1}{2}x</math>.<br/><br/>
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The equation of the line can be written as <math>y= -\tfrac{1}{2}x + 2</math> , and then we see that its slope is <math>k= -\tfrac{1}{2}</math>and that <math>m=2</math>. See the figure below to the right.
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The equation of the line can be written as <math>y= -\tfrac{1}{2}x + 2</math>. From this we see that has the slope <math>k= -\tfrac{1}{2}</math>and that <math>m=2</math>. See the figure below to the right.
</ol>
</ol>
{| align="center" padding="20px"
{| align="center" padding="20px"
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|align="center"|{{:2.2 - Figur - Linjen y = 2x - 1}}
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|align="center"|{{:2.2 - Figure - The line y = 2x - 1}}
|width="20px"|
|width="20px"|
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|align="center"|{{:2.2 - Figur - Linjen y = 2 - x/2}}
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|align="center"|{{:2.2 - Figure - The line y = 2 - x/2}}
|-
|-
|align="center"|<small>Line ''y'' = 2''x'' - 1</small>
|align="center"|<small>Line ''y'' = 2''x'' - 1</small>
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If we plot the points and draw the line in a coordinate system, we see that <math>5-2=3</math> steps in the <math>x</math>-direction corresponds to <math>3-1=2</math> steps in the <math>y</math>-direction along the line. This means that <math>1</math> step in the <math>x</math>-direction corresponds to <math>k=\frac{3-1}{5-2}= \frac{2}{3}</math> steps in the <math>y</math>-direction. So the line's slope is <math>k= \frac{2}{3}</math>.
If we plot the points and draw the line in a coordinate system, we see that <math>5-2=3</math> steps in the <math>x</math>-direction corresponds to <math>3-1=2</math> steps in the <math>y</math>-direction along the line. This means that <math>1</math> step in the <math>x</math>-direction corresponds to <math>k=\frac{3-1}{5-2}= \frac{2}{3}</math> steps in the <math>y</math>-direction. So the line's slope is <math>k= \frac{2}{3}</math>.
-
<center>{{:2.2 - Figur - Linje genom punkterna (2,1) och (5,3)}}</center>
+
<center>python</center>
</div>
</div>
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Two straight lines that are parallel clearly have the same slope. It is also possible to see (such as in the figure below) that for two lines having slopes <math>k_1</math> and <math>k_2</math> and also are perpendicular then <math>k_2 = -\frac{1}{k_1}</math>, which also can be written as <math>k_1 k_2 = -1</math>.
+
Two straight lines that are parallel clearly have the same slope. It is also possible to see (such as in the figure below) that for two lines with slopes <math>k_1</math> and <math>k_2</math> that are perpendicular to one another, then <math>k_2 = -\frac{1}{k_1}</math>, which also can be written as <math>k_1 k_2 = -1</math>.
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<center>{{:2.2 - Figur - Riktningskoefficient för vinkelräta linjer}}</center>
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<center>{{:2.2 - Figure - The slope of perpendicular lines}}</center>
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The straight line in the figure on the left has slope <math>k</math>, that is <math>1</math> step in the <math>x</math>-direction corresponds to <math>k</math> steps in the <math>y</math>-direction. If the line is rotated <math>90^\circ</math> clockwise, we get the line in the figure to the right, and that line has slope <math>-\frac{1}{k}</math> because now <math>-k</math> steps in the <math>x</math>-direction corresponds to <math>1</math> step in the <math>y</math>-direction
+
The straight line in the figure on the left has slope <math>k</math>, that is <math>1</math> step in the <math>x</math>-direction corresponds to <math>k</math> steps in the <math>y</math>-direction. If the line is rotated <math>90^\circ</math> clockwise we get the line in the figure to the right. That line has slope <math>-\frac{1}{k}</math> because now <math>-k</math> steps in the <math>x</math>-direction corresponds to <math>1</math> step in the <math>y</math>-direction
<div class="exempel">
<div class="exempel">
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All straight lines (including vertical lines) can be put into the general form
All straight lines (including vertical lines) can be put into the general form
<div class="regel">
<div class="regel">
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{{Fristående formel||<math>ax+by=c</math>}}
+
{{Displayed math||<math>ax+by=c</math>}}
</div>
</div>
where <math>a</math>, <math>b</math> and <math>c</math> are constants.
where <math>a</math>, <math>b</math> and <math>c</math> are constants.
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Move the <math>x</math>-term to the left-hand side: <math>-5x+y=7</math>.</li>
Move the <math>x</math>-term to the left-hand side: <math>-5x+y=7</math>.</li>
<li> Put the line <math>2x+3y=-1</math> into the form <math>y=kx+m</math>.<br/><br/>
<li> Put the line <math>2x+3y=-1</math> into the form <math>y=kx+m</math>.<br/><br/>
-
Move the <math>x</math>-term to the right-hand side <math>3y=-2x-1 </math>and divide both sides by <math>3</math>
+
Move the <math>x</math>-term to the right-hand side <math>3y=-2x-1 </math> and divide both sides by <math>3</math>
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{{Fristående formel||<math>y=-\frac{2}{3}x - \frac{1}{3}\,\mbox{.}</math>}}
+
{{Displayed math||<math>y=-\frac{2}{3}x - \frac{1}{3}\,\mbox{.}</math>}}
</ol>
</ol>
</div>
</div>
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==Regions in a coordinate system ==
==Regions in a coordinate system ==
-
By geometrically interpreting inequalities, one can describe regions in the plane.
+
By geometrically interpreting inequalities one can describe regions in the plane.
<div class="exempel">
<div class="exempel">
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<li>Sketch the region in the <math>x,y</math>-plane that satisfies <math>y\ge2</math>. <br/><br/>
<li>Sketch the region in the <math>x,y</math>-plane that satisfies <math>y\ge2</math>. <br/><br/>
The region is given by all the points <math>(x,y)</math> for which the <math>y</math>-coordinate is equal or greater than <math>2</math> that is all points on or above the line <math>y=2</math>.<br/>
The region is given by all the points <math>(x,y)</math> for which the <math>y</math>-coordinate is equal or greater than <math>2</math> that is all points on or above the line <math>y=2</math>.<br/>
-
<center>{{:2.2 - Figur - Området y ≥ 2}}</center></li>
+
<center>{{:2.2 - Figure - The region y ≥ 2}}</center></li>
<li>Sketch the region in the <math>x,y</math>-plane that satisfies <math>y < x</math>. <br/><br/>
<li>Sketch the region in the <math>x,y</math>-plane that satisfies <math>y < x</math>. <br/><br/>
A point <math>(x,y)</math> that satisfies the inequality <math>y < x</math> must have an <math>x</math>-coordinate that is larger than its <math>y</math>-coordinate. Thus the area consists of all the points to the right of the line <math>y=x</math>.<br/>
A point <math>(x,y)</math> that satisfies the inequality <math>y < x</math> must have an <math>x</math>-coordinate that is larger than its <math>y</math>-coordinate. Thus the area consists of all the points to the right of the line <math>y=x</math>.<br/>
-
<center>{{:2.2 - Figur - Området y mindre än x}}</center>
+
<center>{{:2.2 - Figure - The region y less than x}}</center>
The fact that the line <math>y=x</math> is dashed means that the points on the line do not belong to the coloured area.
The fact that the line <math>y=x</math> is dashed means that the points on the line do not belong to the coloured area.
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The double inequality can be divided into two inequalities
The double inequality can be divided into two inequalities
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{{Fristående formel||<math>3x+2y \ge 2 \quad</math> and <math>\quad 3x+2y\le4 \;\mbox{.}</math>}}
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{{Displayed math||<math>3x+2y \ge 2 \quad</math> and <math>\quad 3x+2y\le4 \;\mbox{.}</math>}}
We move the <math>x</math>-terms into the right-hand side and divide both sides by <math>2</math> giving
We move the <math>x</math>-terms into the right-hand side and divide both sides by <math>2</math> giving
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{{Fristående formel||<math>y \ge 1-\frac{3}{2}x \quad</math> and <math>\quad y\le 2-\frac{3}{2}x \;\mbox{.}</math>}}
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{{Displayed math||<math>y \ge 1-\frac{3}{2}x \quad</math> and <math>\quad y\le 2-\frac{3}{2}x \;\mbox{.}</math>}}
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The points that satisfy the first inequality are on and above the line <math>y \ge 1-\tfrac{3}{2}x</math> while the points that satisfy the other inequality are on or below the line <math>y\le 2-\tfrac{3}{2}x</math>.
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The points that satisfy the first inequality are on and above the line <math>y = 1-\tfrac{3}{2}x</math> while the points that satisfy the other inequality are on or below the line <math>y = 2-\tfrac{3}{2}x</math>.
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<center>{{:2.2 - Figur - Områdena 3x + 2y ≥ 2 och 3x + 2y ≤ 4}}</center>
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<center>{{:2.2 - Figure - The regions 3x + 2y ≥ 2 and 3x + 2y ≤ 4}}</center>
<center><small>The figure on the left shows the region <math>3x+2y\ge 2</math> and figure to the right shows the region <math>3x+2y\le 4</math>.</small></center>
<center><small>The figure on the left shows the region <math>3x+2y\ge 2</math> and figure to the right shows the region <math>3x+2y\le 4</math>.</small></center>
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Points that satisfy both inequalities form a band-like region where both coloured areas overlap.
Points that satisfy both inequalities form a band-like region where both coloured areas overlap.
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<center>{{:2.2 - Figur - Området 2 ≤ 3x + 2y ≤ 4}}</center>
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<center>{{:2.2 - Figure - The region 2 ≤ 3x + 2y ≤ 4}}</center>
<center><small>The figure shows the region <math>2\le 3x+2y\le 4</math>.</small></center>
<center><small>The figure shows the region <math>2\le 3x+2y\le 4</math>.</small></center>
</div>
</div>
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If we draw the lines <math>y=x</math>, <math>y=-x</math> and <math>y=2</math> then these lines bound a triangle in a coordinate system.
If we draw the lines <math>y=x</math>, <math>y=-x</math> and <math>y=2</math> then these lines bound a triangle in a coordinate system.
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<center>{{:2.2 - Figur - Triangel begränsad av y = x, y = 2 och y = -x}}</center>
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<center>{{:2.2 - Figure - The triangle bounded by y = x, y = 2 and y = -x}}</center>
We find that for a point to lie in this triangle, it has to satisfy certain conditions.
We find that for a point to lie in this triangle, it has to satisfy certain conditions.
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We see that its <math>y</math>-coordinate must be less than <math>2</math>. At the same time, we see that the triangle is bounded by <math> y=0</math> below. Thus the <math>y</math> coordinates must be in the range <math> 0\le y\le2</math>.
We see that its <math>y</math>-coordinate must be less than <math>2</math>. At the same time, we see that the triangle is bounded by <math> y=0</math> below. Thus the <math>y</math> coordinates must be in the range <math> 0\le y\le2</math>.
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For the <math>x</math>-coordinate, the situation is a little more complicated.
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For the <math>x</math>-coordinate the situation is a little more complicated.
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We see that the <math>x</math>-coordinate must satisfy the fact that all points lie above the lines <math>y=-x</math> and <math>y=x</math>. We see that this is satisfied if <math>-y\le x\le y</math>. Since we already have restricted the <math>y</math>-coordinates, we see that <math>x</math> cannot be larger than <math>2</math>
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We see that the <math>x</math>-coordinate must satisfy the fact that all points lie above the lines <math>y=-x</math> and <math>y=x</math>. We see that this is satisfied if <math>-y\le x\le y</math>. Since we already have restricted the <math>y</math>-coordinates we find that <math>x</math> cannot be larger than <math>2</math>
or less than <math>-2</math>.
or less than <math>-2</math>.
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Thus the base of the triangle is <math>4</math> units of length and the height <math>2</math> units of length.
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This gives the base of the triangle as <math>4</math> units of length and a height of <math>2</math> units of length.
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The area of this triangle is therefore <math> 4\cdot 2/2=4</math> units of area.
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The area of this triangle is therefore <math> 4\cdot 2/2=4</math> square units.
</div>
</div>
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'''Basic and final tests'''
'''Basic and final tests'''
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After you have read the text and worked through the exercises, you should do the basic and final tests to pass this section. You can find the link to the tests in your student lounge.
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After you have read the text and worked through the exercises you should do the basic and final tests to pass this section. You can find the link to the tests in your student lounge.
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'''Keep in mind that ... '''
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'''Keep in mind that... '''
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Draw your own diagrams when you solve geometrical problems and draw carefully and accurately! A good diagram can mean you are halfway to a solution, but a poor diagram may well fool you.
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You should draw your own diagrams when you solve geometrical problems and draw them carefully and accurately! A good diagram can mean you are halfway to a solution, however a poor diagram may well fool you.
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'''Reviews'''
'''Reviews'''
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For those of you who want to deepen your studies or need more detailed explanations consider the following references:
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For those of you who want to deepen your studies or need a more detailed explanation, consider the following references:
[http://matmin.kevius.com/linje.html Learn more about linear equations in Bruno Kevius mathematical glossary (Swedish).]
[http://matmin.kevius.com/linje.html Learn more about linear equations in Bruno Kevius mathematical glossary (Swedish).]

Current revision

       Theory          Exercises      

Contents:

  • First degree equations
  • Equation of a straight line
  • Geometrical problems
  • Regions that are defined using inequalities

Learning outcomes:

After this section you will have learned how to:

  • Solve algebraic equations, which after simplification results in first degree equations.
  • Convert between the forms y = kx + m and ax + by + c = 0.
  • Sketch straight lines from their equation.
  • Solve geometric problems that contain straight lines.
  • Sketch regions defined by linear inequalities and determine the area of these regions.

First degree equations

To solve first degree equations (also known as linear equations) we perform calculations on both sides simultaneously. This gradually simplifies the equation and ultimately leads to \displaystyle x being alone on one side of the equation.

Example 1

  1. Solve the equation \displaystyle x+3=7.

    Subtract \displaystyle 3 from both sides
    \displaystyle x+3-3=7-3.
    The left-hand side then simplifies to \displaystyle x and we get
    \displaystyle x=7-3=4.
  2. Solve the equation \displaystyle 3x=6.

    Divide both sides by \displaystyle 3
    \displaystyle \frac{3x}{3} = \frac{6}{3}\,.
    After having cancelled \displaystyle 3 on the left-hand side we have
    \displaystyle x=\frac{6}{3} = 2.
  3. Solve the equation \displaystyle 2x+1=5\,\mbox{.}

    First we subtract \displaystyle 1 from both sides to get \displaystyle 2x on its own on the left-hand side
    \displaystyle 2x=5-1.
    We then divide both sides by \displaystyle 2 to get the answer
    \displaystyle x = \frac{4}{2} = 2.

A first degree equation can be written in the normal form \displaystyle ax=b. The solution is then simply \displaystyle x=b/a (we must assume that \displaystyle a\not=0). The possible difficulties that may occur when you solve a first degree equation are thus not in themselves the form of the solution, but rather the simplifications that may be needed to achieve the normal form. Below are a couple examples in which the equation can be simplified to a linear normal form, thus having a unique solution.

Example 2

Solve the equation\displaystyle \,2x-3=5x+7.


Since \displaystyle x occurs on both the left and right hand sides we subtract \displaystyle 2x from both sides

\displaystyle 2x-3-2x=5x+7-2x

and now \displaystyle x only appears on the right-hand side

\displaystyle -3 = 3x+7 \; \mbox{.}

We now subtract 7 from both sides

\displaystyle -3 -7 = 3x +7-7

and get \displaystyle 3x on its own on the right-hand side

\displaystyle -10=3x\,\mbox{.}

The last step is to divide both sides by \displaystyle 3

\displaystyle \frac{-10}{3} = \frac{3x}{3}

giving

\displaystyle x=-\frac{10}{3}\,\mbox{.}

Example 3

Solve for \displaystyle x the equation \displaystyle ax+7=3x-b.


By subtracting \displaystyle 3x from both sides

\displaystyle ax+7-3x=3x-b-3x
\displaystyle ax+7-3x=\phantom{3x}{}-b\phantom{{}-3x}

and then subtracting \displaystyle 7

\displaystyle ax+7-3x -7=-b-7
\displaystyle ax\phantom{{}+7}{}-3x\phantom{{}-7}{}=-b-7

we have gathered together all the terms that contain \displaystyle x on the left-hand side and all other terms on the right-hand side. Since the terms on the left-hand side have \displaystyle x as a common factor \displaystyle x, they can be factored out

\displaystyle (a-3)x = -b-7\; \mbox{.}

Divide both sides with \displaystyle a-3 giving

\displaystyle x= \frac{-b-7}{a-3}\; \mbox{.}

It is not always obvious that we are dealing with a first degree equation. In the following two examples simplifications turn the original equation into a first degree equation.

Example 4

Solve the equation\displaystyle \ (x-3)^2+3x^2=(2x+7)^2.


Expand the quadratic expressions on both sides

\displaystyle x^2-6x+9+3x^2=4x^2+28x+49\,\mbox{,}
\displaystyle 4x^2-6x+9=4x^2+28x+49\,\mbox{.}

Subtract \displaystyle 4x^2 from both sides

\displaystyle -6x +9 = 28x +49\; \mbox{.}

Add \displaystyle 6x to both sides

\displaystyle 9 = 34x +49\; \mbox{.}

Subtract \displaystyle 49 from both sides

\displaystyle -40=34x\; \mbox{.}

Divide both sides by \displaystyle 34

\displaystyle x = \frac{-40}{34}= - \frac{20}{17}\; \mbox{.}

Example 5

Solve the equation \displaystyle \ \frac{x+2}{x^2+x} = \frac{3}{2+3x}.


Collect both terms to one side

\displaystyle \frac{x+2}{x^2+x}-\frac{3}{2+3x}= 0\; \mbox{.}

Convert the terms so that they have the same denominator

\displaystyle \frac{(x+2)(2+3x)}{(x^2+x)(2+3x)}-\frac{3(x^2+x)}{(2+3x)(x^2+x)}= 0

and simplify the numerator

\displaystyle \frac{(x+2)(2+3x)-3(x^2+x)}{(x^2+x)(2+3x)} = 0,
\displaystyle \frac{3x^2+8x+4-(3x^2+3x)}{(x^2+x)(2+3x)} = 0,
\displaystyle \frac{5x +4}{(x^2+x)(2+3x)} = 0\,\mbox{.}

This equation is only satisfied when the numerator is equal to zero (whilst the denominator is not equal to zero);

\displaystyle 5x+4=0

which gives that \displaystyle \,x = -\frac{4}{5}.


Straight lines

Functions such as

\displaystyle y = 2x+1
\displaystyle y = -x+3
\displaystyle y = \frac{1}{2} x -5

are examples of linear functions. They have the general form

\displaystyle y = kx+m

where \displaystyle k and \displaystyle m are constants.

The graph of a linear function is always a straight line. The constant \displaystyle k indicates the slope of the line with respect to the \displaystyle x-axis and \displaystyle m gives the \displaystyle y-coordinate of the point where the line intersects the \displaystyle y-axis.

[Image]

The line y = kx + m has slope k and cuts the y-axis at (0,m)

The constant \displaystyle k is called the slope and states that given a unit change in the positive \displaystyle x-direction, along the line there is a \displaystyle k unit change in the positive \displaystyle y-direction. Thus if

  • \displaystyle k>0\, the line slopes upwards,
  • \displaystyle k<0\, the line slopes downwards.

For a horizontal line (parallel to the \displaystyle x-axis) \displaystyle k=0, whereas a vertical line (parallel to the \displaystyle y-axis) does not have a \displaystyle k value ( a vertical line cannot be written in the form \displaystyle y=kx+m).

Example 6

  1. Sketch the line \displaystyle y=2x-1.

    Comparing with the standard equation \displaystyle y=kx+m, we see that \displaystyle k=2 and \displaystyle m=-1. This gives the line's slope as \displaystyle 2 and that it cuts the \displaystyle y-axis at \displaystyle (0,-1). See the figure below-left.
  2. Sketch the line \displaystyle y=2-\tfrac{1}{2}x.

    The equation of the line can be written as \displaystyle y= -\tfrac{1}{2}x + 2. From this we see that has the slope \displaystyle k= -\tfrac{1}{2}and that \displaystyle m=2. See the figure below to the right.

[Image]

[Image]

Line y = 2x - 1 Line y = 2 - x/2

Example 7

What is the slope of the straight line that passes through the points \displaystyle (2,1) and \displaystyle (5,3)?


If we plot the points and draw the line in a coordinate system, we see that \displaystyle 5-2=3 steps in the \displaystyle x-direction corresponds to \displaystyle 3-1=2 steps in the \displaystyle y-direction along the line. This means that \displaystyle 1 step in the \displaystyle x-direction corresponds to \displaystyle k=\frac{3-1}{5-2}= \frac{2}{3} steps in the \displaystyle y-direction. So the line's slope is \displaystyle k= \frac{2}{3}.

python

Two straight lines that are parallel clearly have the same slope. It is also possible to see (such as in the figure below) that for two lines with slopes \displaystyle k_1 and \displaystyle k_2 that are perpendicular to one another, then \displaystyle k_2 = -\frac{1}{k_1}, which also can be written as \displaystyle k_1 k_2 = -1.

[Image]

The straight line in the figure on the left has slope \displaystyle k, that is \displaystyle 1 step in the \displaystyle x-direction corresponds to \displaystyle k steps in the \displaystyle y-direction. If the line is rotated \displaystyle 90^\circ clockwise we get the line in the figure to the right. That line has slope \displaystyle -\frac{1}{k} because now \displaystyle -k steps in the \displaystyle x-direction corresponds to \displaystyle 1 step in the \displaystyle y-direction

Example 8

  1. The lines \displaystyle y=3x-1 and \displaystyle y=3x+5 are parallel.
  2. The lines \displaystyle y=x+1 and \displaystyle y=2-x are perpendicular.

All straight lines (including vertical lines) can be put into the general form

\displaystyle ax+by=c

where \displaystyle a, \displaystyle b and \displaystyle c are constants.

Example 9

  1. Put the line \displaystyle y=5x+7 into the form \displaystyle ax+by=c.

    Move the \displaystyle x-term to the left-hand side: \displaystyle -5x+y=7.
  2. Put the line \displaystyle 2x+3y=-1 into the form \displaystyle y=kx+m.

    Move the \displaystyle x-term to the right-hand side \displaystyle 3y=-2x-1 and divide both sides by \displaystyle 3
    \displaystyle y=-\frac{2}{3}x - \frac{1}{3}\,\mbox{.}

Here you can see how an equation for a line can be obtained if we know the coordinates of two points on the line.

Here you can vary k and m and see how this affects the line's characteristics.


Regions in a coordinate system

By geometrically interpreting inequalities one can describe regions in the plane.

Example 10

  1. Sketch the region in the \displaystyle x,y-plane that satisfies \displaystyle y\ge2.

    The region is given by all the points \displaystyle (x,y) for which the \displaystyle y-coordinate is equal or greater than \displaystyle 2 that is all points on or above the line \displaystyle y=2.

    [Image]

  2. Sketch the region in the \displaystyle x,y-plane that satisfies \displaystyle y < x.

    A point \displaystyle (x,y) that satisfies the inequality \displaystyle y < x must have an \displaystyle x-coordinate that is larger than its \displaystyle y-coordinate. Thus the area consists of all the points to the right of the line \displaystyle y=x.

    [Image]

    The fact that the line \displaystyle y=x is dashed means that the points on the line do not belong to the coloured area.

Example 11

Sketch the region in the \displaystyle x,y-plane that satisfies \displaystyle 2 \le 3x+2y\le 4.


The double inequality can be divided into two inequalities

\displaystyle 3x+2y \ge 2 \quad and \displaystyle \quad 3x+2y\le4 \;\mbox{.}

We move the \displaystyle x-terms into the right-hand side and divide both sides by \displaystyle 2 giving

\displaystyle y \ge 1-\frac{3}{2}x \quad and \displaystyle \quad y\le 2-\frac{3}{2}x \;\mbox{.}

The points that satisfy the first inequality are on and above the line \displaystyle y = 1-\tfrac{3}{2}x while the points that satisfy the other inequality are on or below the line \displaystyle y = 2-\tfrac{3}{2}x.

[Image]

The figure on the left shows the region \displaystyle 3x+2y\ge 2 and figure to the right shows the region \displaystyle 3x+2y\le 4.


Points that satisfy both inequalities form a band-like region where both coloured areas overlap.

[Image]

The figure shows the region \displaystyle 2\le 3x+2y\le 4.

Example 12

If we draw the lines \displaystyle y=x, \displaystyle y=-x and \displaystyle y=2 then these lines bound a triangle in a coordinate system.

[Image]

We find that for a point to lie in this triangle, it has to satisfy certain conditions.

We see that its \displaystyle y-coordinate must be less than \displaystyle 2. At the same time, we see that the triangle is bounded by \displaystyle y=0 below. Thus the \displaystyle y coordinates must be in the range \displaystyle 0\le y\le2.

For the \displaystyle x-coordinate the situation is a little more complicated. We see that the \displaystyle x-coordinate must satisfy the fact that all points lie above the lines \displaystyle y=-x and \displaystyle y=x. We see that this is satisfied if \displaystyle -y\le x\le y. Since we already have restricted the \displaystyle y-coordinates we find that \displaystyle x cannot be larger than \displaystyle 2 or less than \displaystyle -2.

This gives the base of the triangle as \displaystyle 4 units of length and a height of \displaystyle 2 units of length.

The area of this triangle is therefore \displaystyle 4\cdot 2/2=4 square units.


Exercises

Study advice

Basic and final tests

After you have read the text and worked through the exercises you should do the basic and final tests to pass this section. You can find the link to the tests in your student lounge.


Keep in mind that...

You should draw your own diagrams when you solve geometrical problems and draw them carefully and accurately! A good diagram can mean you are halfway to a solution, however a poor diagram may well fool you.


Useful web sites

Experiment with Equations of a Straight Line

Experiment with Archimedes Triangle & Squaring of Parabola.

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