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2.2 Linear expressions

From Förberedande kurs i matematik 1

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Two straight lines that are parallel clearly have the same gradient. It is also possible to see (such as in the figure below) that if two lines with gradients <math>k_1</math> and <math>k_2</math> 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>.
Two straight lines that are parallel clearly have the same gradient. It is also possible to see (such as in the figure below) that if two lines with gradients <math>k_1</math> and <math>k_2</math> 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 - Figure - The slope of perpendicular lines}}</center>
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<center>{{:2.2 - Figure - The gradients of perpendicular lines}}</center>
The straight line in the figure on the left has gradient <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 gradient <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 gradient <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 gradient <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

Revision as of 14:48, 6 January 2009

       Theory          Exercises      

Contents:

  • Linear 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 equations that are linear, or linear when simplified.
  • Convert between the forms y = kx + m and ax + by + c = 0.
  • Sketch straight lines from their equations.
  • 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 linear equations (also known as equations of degree one) we perform operations on both sides simultaneously, in such a way as to gradually simplify the equation and ultimately lead to x being alone on one side of it.

Example 1

  1. Solve the equation x+3=7.

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

    Divide both sides by 3
    33x=36.
    After cancelling 3 on the left-hand side we have
    x=36=2.
  3. Solve the equation 2x+1=5.

    First we subtract 1 from both sides to get 2x on its own on the left-hand side
    2x=5−1.
    We then divide both sides by 2 to get the answer
    x=24=2.

A linear equation can always be written in the form ax=b. The solution is then simply x=ba (we must assume that a=0). Sometimes, however, some work is needed in order to place the equation in this form. Here are a couple of examples.

Example 2

Solve the equation2x−3=5x+7.


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

2x−3−2x=5x+7−2x

and now x only appears on the right-hand side

−3=3x+7.

We now subtract 7 from both sides

−3−7=3x+7−7

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

−10=3x.

The last step is to divide both sides by 3

3−10=33x

giving

x=−310.

Example 3

Solve for x the equation ax+7=3x−b.


By subtracting 3x from both sides

ax+7−3x=3x−b−3x
ax+7−3x=−b

and then subtracting 7

ax+7−3x−7=−b−7
ax−3x=−b−7

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

(a−3)x=−b−7.

Divide both sides with a−3 giving

x=a−3−b−7.

It is not always obvious that we are dealing with a linear equation. In the following two examples simplifications are needed to convert the original equation into a linear one.

Example 4

Solve the equation (x−3)2+3x2=(2x+7)2.


Expand the quadratic expressions on both sides

x2−6x+9+3x2=4x2+28x+49,
4x2−6x+9=4x2+28x+49.

Subtract 4x2 from both sides

−6x+9=28x+49.

Add 6x to both sides

9=34x+49.

Subtract 49 from both sides

−40=34x.

Divide both sides by 34

x=34−40=−1720.

Example 5

Solve the equation  x+2x2+x=32+3x.


Collect both terms to one side

x+2x2+x−32+3x=0.

Convert the terms so that they have the same denominator

(x+2)(2+3x)(x2+x)(2+3x)−3(x2+x)(2+3x)(x2+x)=0

and simplify the numerator

(x2+x)(2+3x)(x+2)(2+3x)−3(x2+x)=0
(x2+x)(2+3x)3x2+8x+4−(3x2+3x)=0
5x+4(x2+x)(2+3x)=0.

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

5x+4=0

which gives that x=−54.


Straight lines

Functions such as

y=2x+1
y=−x+3
y=21x−5

are examples of linear functions. They have the general form

y=kx+m

where k and m are constants.

The graph of a linear function is always a straight line. The constant k indicates the gradient of the line with respect to the x-axis and m gives the intercept: that is, the y-coordinate of the point where the line intersects the y-axis.

[Image]

The line y = mx + c has gradient m and cuts the y-axis at (0,c)

The constant k is called the gradient. To state that a straight line has gradient k means that for each increase of one unit in the positive x-direction there is an increase of k units in the positive \displaystyle y-direction. The gradient can thus be described as the "height of the unit step". Note that a negative gradient means that \displaystyle y decreases as \displaystyle x increases: that is, 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 tells us that the line's gradient is \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 gradient \displaystyle k= -\tfrac{1}{2}and that the intercept \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 gradient 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 correspond 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 gradient is \displaystyle k= \frac{2}{3}.

python

Two straight lines that are parallel clearly have the same gradient. It is also possible to see (such as in the figure below) that if two lines with gradients \displaystyle k_1 and \displaystyle k_2 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 gradient \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 gradient \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 to, 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 shown dashed signifies that the points on the line do not belong to the shaded 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 shaded 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 the \displaystyle xy plane.

[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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