Unit 5 Section 3 : Linear Equations

In this section we consider solving linear equations, using both algebra and graphs.

Example 1

Solve the following equations:

(a) x + 6 = 13
x + 6=13
x=13 – 6 (subtracting 6 from both sides)
x=7
(b) x – 7 = 11
x – 7=11
x=11 + 7 (adding 7 to both sides)
x=18
(c) 4x = 72
4x=72
x= (dividing both sides by 4)
x=18
(d)
= 11
=11
x=11 × 3 (multiplying both sides by 3)
x=33

Example 2

Solve the following equations:

(a) 2x + 4 = 20
2x + 4=20
2x=20 – 4 (subtracting 4 from both sides)
2x=16
x= (dividing both sides by 2)
x=8
(b)
= 3
=3
x + 4=3 × 6 (multiplying both sides by 6)
x + 4=18
x=18 – 4 (subtracting 4 from both sides)
x=14
(c) 4(x + 4) = 18
4(x + 4)=18
4x + 16=18 (removing brackets)
4x=18 – 16 (subtracting 16 from both sides)
4x=2
x= (dividing both sides by 4)
x=

Example 3

Solve the following equations:

(a) 4x + 2 = 3x + 5
4x + 2=3x + 5
x + 2=5 (subtracting 3x from both sides)
x=5 – 2 (subtracting 2 from both sides)
x=3
(b) 4x – 4 = 10 – 3x
4x – 4=10 – 3x
7x – 4=10 (adding 3x to both sides)
7x=10 + 4 (adding 4 to both sides)
7x=14
x= (dividing both sides by 7)
x=2

Example 4

Use graphs to solve the following equations:

(a)
4x – 7 = 9

Draw the lines y = 4x – 7 and y = 9.

The solution is given by the value on the x-axis immediately below the point where y = 4x – 7 and y = 9 cross.

The solution is x = 4.

(b)
x + 7 = 3x – 3

Draw the lines y = x + 7 and y = 3x – 3.

The lines cross where x = 5, so this is the solution of the equation.

Exercises

Question 1
Solve the following equations:
(a)x + 6 = 14 x =
(b)x – 3 = 8 x =
(c)7x = 21 x =
(d)
= 10
x =
(e)10x = 80 x =
(f)5x = 35 x =
(g)x + 9 = 22 x =
(h)x – 4 = 3 x =
(i)x – 22 = 18 x =
(j)
= 100
x =
(k)3x = 96 x =
(l)x + 22 = 47 x =
Question 2
Solve the following equations:
(a)2x + 7 = 15 x =
(b)5x – 3 = 32 x =
(c)6x + 4 = 22 x =
(d)11x – 3 = 19 x =
(e)5x + 2 = 37 x =
(f)
= 21
x =
(g)
= 5
x =
(h)4(x + 2) = 28 x =
(i)3(5x – 6) = 147 x =
(j)2(3x – 7) = 46 x =
(k)
= 6
x =
(l)5(2x + 3) = 35 x =
Question 3
Solve the following equations:
(a)x + 1 = 2x – 1 x =
(b)2x + 4 = 3x – 1 x =
(c)7x – 2 = 5x + 6 x =
(d)4x + 7 = 10x – 11 x =
(e)x + 18 = 9x – 22 x =
(f)7x + 1 = 3x + 17 x =
(g)6(x + 1) = 14(x – 1) x =
(h)2(5x + 3) = 12x – 3 x =
Question 4
The graph y = 2x – 5 is shown:
Use the graph to solve the equations:
(a) 2x – 5 = 1x =
(b) 2x – 5 = 7x =
(c) 2x – 5 = –3x =
Question 5
Solve the equation 2x – 3 = 9 by drawing the graphs y = 2x – 3 and y = 9.
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x =
Question 6
Use a graph to solve the equation 4x – 5 = 3.
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x =
Question 7
(a)

On the same set of axes, draw the lines with equations y = x + 1 and y = 2x – 3.

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(b)

Use the graph to find the solution of the equation x + 1 = 2x – 3

x =
Question 8
Use a graph to solve the following equations:
(a)
2x = –x + 3
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x =
(b)
4 – 2x = 2x – 8
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x =
Question 9
The following graph shows the lines with equations y = 2x + 1, y = x + 2 and y = 10 – x.
Use the graph to solve the equations:
(a)2x + 1 = 10 – xx =
(b)x + 2 = 10 – xx =
(c)2x + 1 = x + 2x =
Question 10
On the same set of axes, draw the graphs of three straight lines and use them to solve the equations:
(a)2x – 2 = x + 3
(b)2x – 2 = 8
(c)x + 3 = 8
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(a)x =
(b)x =
(c)x =
Question 11
Solve these equations.
(a)

4 – 2y = 10 – 6y

y =
4 – 2y=10 – 6y
4 + 4y=10(adding 6y to both sides)
4y=6(subtracting 4 from both sides)
y=(dividing both sides by 4)
y=( = 1.5)
(b)

5y + 20 = 3(y – 4)

y =
5y + 20=3(y – 4)
5y + 20=3y – 12(removing brackets)
2y + 20=–12(subtracting 3y from both sides)
2y=–32(subtracting 20 from both sides)
y=–16