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2 Algebra
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Australia
Year 9

2.04 Solving linear equations

Lesson
Worksheet
Practice
Worksheet
One-step equations
1

For each of the following equations, describe what must be done to both sides in order to solve the equation:

a

x - 4 = 6

b
x + 2 = 9
c
4x = 92
d
\dfrac{x}{5} = 7
2

Solve:

a
v + 37 = 8
b
x - 45 = 5
c
-5 j = 75
d
6 k = - 54
e
\dfrac{t}{5} = 0.4
f
\dfrac{s}{7} = 0.5
g
\dfrac{44}{t} = - 4
h
\dfrac{63}{u} = - 9
3

Consider the equation - \dfrac{q}{9} = 6.

a

Describe what must be done to make q the subject of the equation.

b

Hence, find q.

Two-step equations
4

Solve:

a
3x-4=11
b
4 s + 3 = - 25
c
6v - 8 = - 26
d

3 x + 11 = 5

e
4 x - 6 x = 4
f
- 10 + 3 k = 5
g
8 m + 9 = 65
h

12 q + 8 = 26

i

4 s - 9 = 39

j

- 6 s + 5 = 47

k

6 t + 2 = - 22

l
- w - 7 = 7
m
5 w - 8 = 2
n
- 63 - 9 y = 63
Equations with brackets
5

Ryan attempted to solve the equation 9 \left( 4x - 6 \right) = 18. His working is shown below:

\begin{aligned} 9 \left( 4x - 6 \right) &= 18 \\ 4x - 6 &= 9 \\ 4x &= 15 \\x &= \dfrac{15}{4} \end{aligned}
a

What was his mistake?

b

Solve the equation correctly.

6

Solve:

a
2 \left( 3 x + 1\right) = 14
b
3 \left(v + 5\right) + 4 = 37
c
- 3 \left( 3 x + 4\right) = 24
d
- 5 \left( 2 x - 5\right) = 75
e
- 8 \left(h + 2\right) = - 88
f
4 \left(x + 6\right) = 41
g
5 \left(x - 6\right) = - 59
h
- 3 \left( 2 x - 6\right) + 5 \left( 3 x - 5\right) = 20
i
6 \left( 5 x - 6\right) - 4 x + 8 = 50
j
4\left(4x-5\right)-3x+8=40
k
2 \left( 2 x + 5\right) + 3 \left( 4 x + 6\right) = 76
l
4 \left( 3 x + 6\right) - 3 \left( 2 x + 5\right) = 27
Equations with pronumerals on both sides
7

Explain what should be done to solve the equation 9 x = 7 + 8 x.

8

Solve:

a
5 x = x + 8
b

7 x = x + 30

c

3 x = x - 18

d
4 x = - 4 x - 32
e

7 x + 6 = 3 x + 22

f

6 x - 3 = 4 x + 7

g

5 x - 3 = - 4 x + 33

h

4 x - 9 = 5 x - 6

9

Solve:

a

5 \left( 2 x + 5\right) =2x + 13

b

6 \left( 3 x + 1\right) =4x + 20

c

4 \left( 3 x - 2\right) =6x + 16

d

2 \left(8 - 3 x\right) =10x - 20

e

3 x + 4 = 5 \left( 6 x + 5\right) + 60

f
5 x + 3 = 8 \left( 6 x + 4\right) + 143
g
3x+6=3\left(8x+4\right)+99
h
8 x - 6 = 4 x + 10
Equations with fractions
10

Solve:

a
\dfrac{2\left(x - 4 \right)}{3}=8
b
\dfrac{5\left(x +1 \right)}{2}=10
c
\dfrac{-3\left(x -3 \right)}{4}=0
d
\dfrac{6\left(x +5 \right)}{7}=12
e
\dfrac{9\left(x +10 \right)}{2}=-6
f
\dfrac{4\left(x - 2 \right)}{3}=-3
g
\dfrac{7\left(x - 9 \right)}{5}=14
h
\dfrac{2\left(x +11 \right)}{9}=8
11

Solve:

a
\dfrac{p - 14}{6} = - 8
b
\dfrac{w}{8} + 17 = 8
c
\dfrac{x}{5}-13=-6
d
\dfrac{t - 6}{2} = 8.5
e
\dfrac{s-8}{4}=3.75
f
\dfrac{7 x}{4} + 14 = 21
g
\dfrac{5y}{4} - 15 = -20
h
\dfrac{x - 8}{5} + 3 = 8
i
\dfrac{z-5}{4}+3=-8
j
\dfrac{- 8 t - 12}{3} = - 12
k
\dfrac{4 x - 70}{6} = 3 x
l
\dfrac{4x-8}{8}=6x-12
m
\dfrac{3 x + 8}{7} = \dfrac{- 3 x + 12}{3}
n
\dfrac{3 x - 5}{7} = - 8 x + 33
o
\dfrac{5x+4}{7}=\dfrac{-3x+12}{3}
Applications
12

Write an equation for each of the following statements, then solve it to find x:

a

When 4 is added to x, the result is 11.

b

When 8 is subtracted from 2x, the result is 6 more than x.

c

When triple the value of x+1 is added to 4, the result is 5 less than the value of 6x.

13

If the perimeter of this triangle is 263\text{ cm}, find the value of x.

14

The following rectangle has a perimeter of 126 + 3 y centimetres.

Find the value of y.

15

A square has a side length of \, 4x + 5\text{ cm}. If the perimeter of the square is 44 \text{ cm}, find the value of x.

16

A rectangle has a width of \, 5x-3\text{ cm} and a height of \, 3x + 7\text{ cm}. If the perimeter of the rectangle is 42 \text{ cm}, find the value of x.

17

A number is multiplied by 5 and then 2 is added. Then the result is multiplied by 6. This is equal to 10 times the number minus 8.

a

Form an equation for this problem.

b

Solve the equation to find the number.

18

A rectangle with a height of 3x + 7 \text{ cm} and a width of 4x has the same perimeter as a square with side length 2x+9\text{ cm}.

Find the value of x.

19

A square with side length 4x - 11 \text{ cm} has the same perimeter as a square with side length 2x+9\text{ cm}.

Find the value of x.

20

Vanessa is cutting out a rectangular board to construct a bookshelf. The board is to have a perimeter of 48 cm, and its length is to be 3 cm shorter than double the width. Let x be the width of the board.

a

Solve for x, the width of the board.

b

Hence, state the length of the board.

21

Valentina tries to guess how many people are at a concert, but she guesses 400 too many. Kenneth guesses 150 too few. The average of their guesses is 3625.

Let x be the exact number of people at the concert. Find the value of x.

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ACMNA215

Sketch linear graphs using the coordinates of two points and solve linear equations

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