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2 Algebra
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AustraliaVIC
Level 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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Outcomes

VCMNA301

Solve problems involving direct proportion. Explore the relationship between graphs and equations corresponding to simple rate problems.

VCMNA310

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

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