2.2 Exercises

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{{Mall:Ej vald flik|[[2.2 Linjära uttryck|Teori]]}}
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{{Not selected tab|[[2.2 Linear expressions|Theory]]}}
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{{Mall:Vald flik|[[2.2 Övningar|Övningar]]}}
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{{Selected tab|[[2.2 Exercises|Exercises]]}}
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===Övning 2.2:1===
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===Exercise 2.2:1===
<div class="ovning">
<div class="ovning">
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L&ouml;s ekvationerna
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Solve the equations
{| width="100%" cellspacing="10px"
{| width="100%" cellspacing="10px"
|a)
|a)
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|| <math>5x+7=2x-6</math>
|| <math>5x+7=2x-6</math>
|}
|}
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</div>{{#NAVCONTENT:Svar|Svar 2.2:1|Lösning a|Lösning 2.2:1a|Lösning b|Lösning 2.2:1b|Lösning c|Lösning 2.2:1c|Lösning d|Lösning 2.2:1d}}
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</div>{{#NAVCONTENT:Answer|Answer 2.2:1|Solution a|Solution 2.2:1a|Solution b|Solution 2.2:1b|Solution c|Solution 2.2:1c|Solution d|Solution 2.2:1d}}
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===Övning 2.2:2===
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===Exercise 2.2:2===
<div class="ovning">
<div class="ovning">
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L&ouml;s ekvationerna
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Solve the equations
{| width="100%" cellspacing="10px"
{| width="100%" cellspacing="10px"
|a)
|a)
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|| <math>(x^2+4x+1)^2+3x^4-2x^2=(2x^2+2x+3)^2</math>
|| <math>(x^2+4x+1)^2+3x^4-2x^2=(2x^2+2x+3)^2</math>
|}
|}
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</div>{{#NAVCONTENT:Svar|Svar 2.2:2|Lösning a|Lösning 2.2:2a|Lösning b|Lösning 2.2:2b|Lösning c|Lösning 2.2:2c|Lösning d|Lösning 2.2:2d}}
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</div>{{#NAVCONTENT:Answer|Answer 2.2:2|Solution a|Solution 2.2:2a|Solution b|Solution 2.2:2b|Solution c|Solution 2.2:2c|Solution d|Solution 2.2:2d}}
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===Övning 2.2:3===
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===Exercise 2.2:3===
<div class="ovning">
<div class="ovning">
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L&ouml;s ekvationerna
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Solve the equations
{| width="100%" cellspacing="10px"
{| width="100%" cellspacing="10px"
|a)
|a)
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|| <math>\left(\displaystyle\frac{2}{x}-3\right)\left(\displaystyle\frac{1}{4x}+\frac{1}{2}\right)-\left(\displaystyle\frac{1}{2x}-\frac{2}{3}\right)^2-\left(\displaystyle\frac{1}{2x}+\frac{1}{3}\right)\left(\displaystyle\frac{1}{2x}-\frac{1}{3}\right)=0</math>
|| <math>\left(\displaystyle\frac{2}{x}-3\right)\left(\displaystyle\frac{1}{4x}+\frac{1}{2}\right)-\left(\displaystyle\frac{1}{2x}-\frac{2}{3}\right)^2-\left(\displaystyle\frac{1}{2x}+\frac{1}{3}\right)\left(\displaystyle\frac{1}{2x}-\frac{1}{3}\right)=0</math>
|}
|}
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</div>{{#NAVCONTENT:Svar|Svar 2.2:3|Lösning a|Lösning 2.2:3a|Lösning b|Lösning 2.2:3b|Lösning c|Lösning 2.2:3c|Lösning d|Lösning 2.2:3d}}
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</div>{{#NAVCONTENT:Answer|Answer 2.2:3|Solution a|Solution 2.2:3a|Solution b|Solution 2.2:3b|Solution c|Solution 2.2:3c|Solution d|Solution 2.2:3d}}
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===Övning 2.2:4===
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===Exercise 2.2:4===
<div class="ovning">
<div class="ovning">
{| width="100%" cellspacing="10px"
{| width="100%" cellspacing="10px"
|a)
|a)
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|width="100%" | Skriv ekvationen f&ouml;r linjen<math>\,y=2x+3\,</math> på formen <math>\,ax+by=c\,</math>
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|width="100%" | Write the equation for the line <math>\,y=2x+3\,</math> in the form <math>\,ax+by=c\,</math>.
|-
|-
|b)
|b)
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|| Skriv ekvationen f&ouml;r linjen<math>,3x+4y-5=0</math> på formen <math>\,y=kx+m\,</math>
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|| Write the equation for the line <math> 3x+4y-5=0</math> in the form <math>\,y=kx+m\,</math>.
|}
|}
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</div>{{#NAVCONTENT:Svar|Svar 2.2:4|Lösning a|Lösning 2.2:4a|Lösning b|Lösning 2.2:4b}}
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</div>{{#NAVCONTENT:Answer|Answer 2.2:4|Solution a|Solution 2.2:4a|Solution b|Solution 2.2:4b}}
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===Övning 2.2:5===
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===Exercise 2.2:5===
<div class="ovning">
<div class="ovning">
{| width="100%" cellspacing="10px"
{| width="100%" cellspacing="10px"
|a)
|a)
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|width="100%" | Best&auml;m ekvationen f&ouml;r den r&auml;ta linje som g&aring;r genom punkterna<math>\,(2,3)\,</math> och <math>\,(3,0)\,</math>
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|width="100%" | Determine the equation for the straight line that goes between the points <math>\,(2,3)\,</math> and<math>\,(3,0)\,</math>.
|-
|-
|b)
|b)
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|| Best&auml;m ekvationen f&ouml;r den r&auml;ta linje som har riktningskoefficient<math>\,-3\,</math> och g&aring;r genom punkten <math>\,(1,-2)\,</math>
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|| Determine the equation for the straight line that has slope <math>\,-3\,</math> and goes through the point <math>\,(1,-2)\,</math>.
|-
|-
|c)
|c)
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|| Best&auml;m ekvationen f&ouml;r den r&auml;ta linje som g&aring;r genom punkten <math>\,(-1,2)\,</math> och &auml;r parallell med linjen <math>\,y=3x+1\,</math>
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|| Determine the equation for the straight line that goes through the point <math>\,(-1,2)\,</math> and is parallel to the line <math>\,y=3x+1\,</math>.
|-
|-
|d)
|d)
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||Best&auml;m ekvationen f&ouml;r den r&auml;ta linje som g&aring;r genom punkten <math>\,(2,4)\,</math> och &auml;r vinkelr&auml;t mot linjen <math>\,y=2x+5\,</math>
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||Determine the equation for the straight line that goes through the point <math>\,(2,4)\,</math> and is perpendicular to the line <math>\,y=2x+5\,</math>.
|-
|-
|e)
|e)
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|| Best&auml;m riktningskoefficienten, <math>\,k\,</math> f&ouml;r den r&auml;ta linje som sk&auml;r ''x''-axeln i punkten <math>\,(5,0)\,</math> och ''y''-axeln i punkten <math>\,(0,-8)\,</math>
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|| Determine the slope, <math>\,k\,</math>, for the straight line that cuts the ''x''-axis at the point <math>\,(5,0)\,</math> and ''y''-axis at the point <math>\,(0,-8)\,</math>.
|}
|}
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</div>{{#NAVCONTENT:Svar|Svar 2.2:5|Lösning a|Lösning 2.2:5a|Lösning b|Lösning 2.2:5b|Lösning c|Lösning 2.2:5c|Lösning d|Lösning 2.2:5d|Lösning e|Lösning 2.2:5e}}
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</div>{{#NAVCONTENT:Answer|Answer 2.2:5|Solution a|Solution 2.2:5a|Solution b|Solution 2.2:5b|Solution c|Solution 2.2:5c|Solution d|Solution 2.2:5d|Solution e|Solution 2.2:5e}}
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===Övning 2.2:6===
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===Exercise 2.2:6===
<div class="ovning">
<div class="ovning">
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Finn sk&auml;rningspunkten mellan f&ouml;ljande linjer
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Find the points of intersection between the pairs of lines in the following
{| width="100%" cellspacing="10px"
{| width="100%" cellspacing="10px"
|a)
|a)
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|width="50%" | <math>y=3x+5\ </math> och ''x''-axeln
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|width="50%" | <math>y=3x+5\ </math> and the ''x''-axis
|b)
|b)
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|width="50%" | <math>y=-x+5\ </math> och ''y''-axeln
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|width="50%" | <math>y=-x+5\ </math> and the ''y''-axis
|-
|-
|c)
|c)
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|width="50%" | <math>4x+5y+6=0\ </math> och ''y''-axeln
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|width="50%" | <math>4x+5y+6=0\ </math> and the ''y''-axis
|d)
|d)
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|| <math>x+y+1=0\ </math> och <math>\ x=12</math>
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|| <math>x+y+1=0\ </math> and <math>\ x=12</math>
|-
|-
|e)
|e)
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|| <math>2x+y-1=0\ </math> och <math>\ y-2x-2=0</math>
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|| <math>2x+y-1=0\ </math> and <math>\ y-2x-2=0</math>
|}
|}
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</div>{{#NAVCONTENT:Svar|Svar 2.2:6|Lösning a|Lösning 2.2:6a|Lösning b|Lösning 2.2:6b|Lösning c|Lösning 2.2:6c|Lösning d|Lösning 2.2:6d|Lösning e|Lösning 2.2:6e}}
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</div>{{#NAVCONTENT:Answer|Answer 2.2:6|Solution a|Solution 2.2:6a|Solution b|Solution 2.2:6b|Solution c|Solution 2.2:6c|Solution d|Solution 2.2:6d|Solution e|Solution 2.2:6e}}
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 +
===Exercise 2.2:7===
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<div class="ovning">
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Sketch the graph of the functions
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{| width="100%" cellspacing="10px"
 +
|a)
 +
|width="33%" | <math>f(x)=3x-2</math>
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|b)
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|width="33%" | <math>f(x)=2-x</math>
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|c)
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|width="33%" | <math>f(x)=2</math>
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|}
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</div>{{#NAVCONTENT:Answer|Answer 2.2:7|Solution a|Solution 2.2:7a|Solution b|Solution 2.2:7b|Solution c|Solution 2.2:7c}}
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 +
===Exercise 2.2:8===
 +
<div class="ovning">
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In the ''xy''-plane, shade in the section whose coordinates <math>\,(x,y)\,</math> satisfy
 +
{| width="100%" cellspacing="10px"
 +
|a)
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|width="33%" | <math>y \geq x </math>
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|b)
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|width="33%" | <math>y &lt; 3x -4 </math>
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|c)
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|width="33%" | <math>2x+3y \leq 6 </math>
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|}
 +
</div>{{#NAVCONTENT:Answer|Answer 2.2:8|Solution a|Solution 2.2:8a|Solution b|Solution 2.2:8b|Solution c|Solution 2.2:8c}}
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 +
===Exercise 2.2:9===
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<div class="ovning">
 +
Calculate the area of the triangle which
 +
{| width="100%" cellspacing="10px"
 +
|a)
 +
|width="100%" | has corners at the points <math>\,(1,4)\,</math>, <math>\,(3,3)\,</math> and <math>\,(1,0)\,</math>.
 +
|-
 +
|b)
 +
|| is bordered by the lines <math>\ x=2y\,</math>, <math>\ y=4\ </math> and <math>\ y=10-2x\,</math>.
 +
|-
 +
|c)
 +
|| is described by the inequalities <math>\ x+y \geq -2\,</math>, <math>\ 2x-y \leq 2\ </math> and <math>\ 2y-x \leq 2\,</math>.
 +
|}
 +
</div>{{#NAVCONTENT:Answer|Answer 2.2:9|Solution a|Solution 2.2:9a|Solution b|Solution 2.2:9b|Solution c|Solution 2.2:9c}}

Current revision

       Theory          Exercises      

Exercise 2.2:1

Solve the equations

a) \displaystyle x-2=-1 b) \displaystyle 2x+1=13
c) \displaystyle \displaystyle\frac{1}{3}x-1=x d) \displaystyle 5x+7=2x-6
Answer
Answer 2.2:1
Solution a
Solution 2.2:1a
Solution b
Solution 2.2:1b
Solution c
Solution 2.2:1c
Solution d
Solution 2.2:1d

Exercise 2.2:2

Solve the equations

a) \displaystyle \displaystyle\frac{5x}{6}-\displaystyle\frac{x+2}{9}=\displaystyle\frac{1}{2} b) \displaystyle \displaystyle\frac{8x+3}{7}-\displaystyle\frac{5x-7}{4}=2
c) \displaystyle (x+3)^2-(x-5)^2=6x+4 d) \displaystyle (x^2+4x+1)^2+3x^4-2x^2=(2x^2+2x+3)^2
Answer
Answer 2.2:2
Solution a
Solution 2.2:2a
Solution b
Solution 2.2:2b
Solution c
Solution 2.2:2c
Solution d
Solution 2.2:2d

Exercise 2.2:3

Solve the equations

a) \displaystyle \displaystyle\frac{x+3}{x-3}-\displaystyle\frac{x+5}{x-2}=0
b) \displaystyle \displaystyle\frac{4x}{4x-7}-\displaystyle\frac{1}{2x-3}=1
c) \displaystyle \left(\displaystyle\frac{1}{x-1}-\frac{1}{x+1}\right)\left(x^2+\frac{1}{2}\right)=\displaystyle\frac{6x-1}{3x-3}
d) \displaystyle \left(\displaystyle\frac{2}{x}-3\right)\left(\displaystyle\frac{1}{4x}+\frac{1}{2}\right)-\left(\displaystyle\frac{1}{2x}-\frac{2}{3}\right)^2-\left(\displaystyle\frac{1}{2x}+\frac{1}{3}\right)\left(\displaystyle\frac{1}{2x}-\frac{1}{3}\right)=0
Answer
Answer 2.2:3
Solution a
Solution 2.2:3a
Solution b
Solution 2.2:3b
Solution c
Solution 2.2:3c
Solution d
Solution 2.2:3d

Exercise 2.2:4

a) Write the equation for the line \displaystyle \,y=2x+3\, in the form \displaystyle \,ax+by=c\,.
b) Write the equation for the line \displaystyle 3x+4y-5=0 in the form \displaystyle \,y=kx+m\,.
Answer
Answer 2.2:4
Solution a
Solution 2.2:4a
Solution b
Solution 2.2:4b

Exercise 2.2:5

a) Determine the equation for the straight line that goes between the points \displaystyle \,(2,3)\, and\displaystyle \,(3,0)\,.
b) Determine the equation for the straight line that has slope \displaystyle \,-3\, and goes through the point \displaystyle \,(1,-2)\,.
c) Determine the equation for the straight line that goes through the point \displaystyle \,(-1,2)\, and is parallel to the line \displaystyle \,y=3x+1\,.
d) Determine the equation for the straight line that goes through the point \displaystyle \,(2,4)\, and is perpendicular to the line \displaystyle \,y=2x+5\,.
e) Determine the slope, \displaystyle \,k\,, for the straight line that cuts the x-axis at the point \displaystyle \,(5,0)\, and y-axis at the point \displaystyle \,(0,-8)\,.
Answer
Answer 2.2:5
Solution a
Solution 2.2:5a
Solution b
Solution 2.2:5b
Solution c
Solution 2.2:5c
Solution d
Solution 2.2:5d
Solution e
Solution 2.2:5e

Exercise 2.2:6

Find the points of intersection between the pairs of lines in the following

a) \displaystyle y=3x+5\ and the x-axis b) \displaystyle y=-x+5\ and the y-axis
c) \displaystyle 4x+5y+6=0\ and the y-axis d) \displaystyle x+y+1=0\ and \displaystyle \ x=12
e) \displaystyle 2x+y-1=0\ and \displaystyle \ y-2x-2=0
Answer
Answer 2.2:6
Solution a
Solution 2.2:6a
Solution b
Solution 2.2:6b
Solution c
Solution 2.2:6c
Solution d
Solution 2.2:6d
Solution e
Solution 2.2:6e

Exercise 2.2:7

Sketch the graph of the functions

a) \displaystyle f(x)=3x-2 b) \displaystyle f(x)=2-x c) \displaystyle f(x)=2
Answer
Answer 2.2:7
Solution a
Solution 2.2:7a
Solution b
Solution 2.2:7b
Solution c
Solution 2.2:7c

Exercise 2.2:8

In the xy-plane, shade in the section whose coordinates \displaystyle \,(x,y)\, satisfy

a) \displaystyle y \geq x b) \displaystyle y < 3x -4 c) \displaystyle 2x+3y \leq 6
Answer
Answer 2.2:8
Solution a
Solution 2.2:8a
Solution b
Solution 2.2:8b
Solution c
Solution 2.2:8c

Exercise 2.2:9

Calculate the area of the triangle which

a) has corners at the points \displaystyle \,(1,4)\,, \displaystyle \,(3,3)\, and \displaystyle \,(1,0)\,.
b) is bordered by the lines \displaystyle \ x=2y\,, \displaystyle \ y=4\ and \displaystyle \ y=10-2x\,.
c) is described by the inequalities \displaystyle \ x+y \geq -2\,, \displaystyle \ 2x-y \leq 2\ and \displaystyle \ 2y-x \leq 2\,.
Answer
Answer 2.2:9
Solution a
Solution 2.2:9a
Solution b
Solution 2.2:9b
Solution c
Solution 2.2:9c
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