﻿ Unit 5 Section 3 : Linear Equations

# 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) (b) x + 1 = 2x – 1 x = 2x + 4 = 3x – 1 x = 7x – 2 = 5x + 6 x = 4x + 7 = 10x – 11 x = x + 18 = 9x – 22 x = 7x + 1 = 3x + 17 x = 6(x + 1) = 14(x – 1) x = 2(5x + 3) = 12x – 3 x =
Question 4
The graph y = 2x – 5 is shown:
Use the graph to solve the equations:
(a) (b) 2x – 5 = 1 x = 2x – 5 = 7 x = 2x – 5 = –3 x =
Question 5
Solve the equation 2x – 3 = 9 by drawing the graphs y = 2x – 3 and y = 9.
x =
Question 6
Use a graph to solve the equation 4x – 5 = 3.
x =
Question 7
(a)

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

(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
x =
(b)
4 – 2x = 2x – 8
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) (b) 2x + 1 = 10 – x x = x + 2 = 10 – x x = 2x + 1 = x + 2 x =
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 2x – 2 = 8 x + 3 = 8
(a) x = x = 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