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8. Equations and Inequalities
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CanadaON
Grade 7

8.04 Two-step equations

Lesson
Worksheet
Practice
Worksheet
Flow charts and operations
1

The flow chart shows operations being performed on s:

\begin{array}{ccccc} & {-7} & & {\times 6} & \\ s & \to & ⬚ & \to & ⬚ \end{array}

Complete the flow chart.

2

The flow chart shows operations being performed on t:

\begin{array}{ccccc} & +3 & & \div 5 & \\ t & \to & ⬚ & \to & ⬚ \end{array}

Complete the flow chart.

3

Complete the following flow charts to form new expressions:

a
\begin{array}{ccccc} & {\times 5} & & {+8} & \\ y & \to & ⬚ & \to & ⬚ \\ \\ \end{array}
b
\begin{array}{ccccc} & \div 3 & & -7 & \\ k & \to & ⬚ & \to & ⬚ \\ \\ \end{array}
c
\begin{array}{ccccc} & +15 & & \times 8 & \\ p & \to & ⬚ & \to & ⬚ \\ \\ \end{array}
d
\begin{array}{ccccc} & {+5} & & {\div3} & \\ n & \to & ⬚ & \to & ⬚ \\\\ \end{array}
e
\begin{array}{ccccc} & \times 4 & & +3 & \\ x & \to & ⬚ & \to & ⬚ \\\\ \end{array}
f
\begin{array}{ccccc} & {\div 9} & & {-7} & \\ j & \to & ⬚ & \to & ⬚ \\ \\ \end{array}
g
\begin{array}{ccccc} & +28 & & \times 3 & \\ p & \to & ⬚ & \to & ⬚ \\ \\ \end{array}
h
\begin{array}{ccccc} & -6 & & \times 7 & \\ s & \to & ⬚ & \to & ⬚ \\ \\ \end{array}
Operations
4

Complete the following flow charts to backtrack to the pronumeral in each expression:

a
\begin{array}{ccccc} & -9 & & \div 6 & \\ 6y+9 & \to & ⬚ & \to & ⬚ \\ \\ \end{array}
b
\begin{array}{ccccc} & \div 5 & & -3 & \\ 5 \left(q + 3\right) & \to & ⬚ & \to & ⬚ \\ \\ \end{array}
c
\begin{array}{ccccc} & \div 6 & & +9 & \\ 6 \left(s - 9\right) & \to & ⬚ & \to & ⬚ \\ \\ \end{array}
d
\begin{array}{ccccc} & \times 5 & & -8 & \\ \dfrac{n + 8}{5} & \to & ⬚ & \to & ⬚ \\ \\ \end{array}
e
\begin{array}{ccccc} & +6 & & \times 5 & \\ \dfrac{j}{5}-6 & \to & ⬚ & \to & ⬚ \\ \\ \end{array}
f
\begin{array}{ccccc} & -9 & & \div 8 & \\ 8y+9 & \to & ⬚ & \to & ⬚ \\ \\ \end{array}
5

Complete the following flow charts by identifying the required operations:

a
\begin{array}{ccccc} & ⬚ & & ⬚ & \\ 4 \left(q + 9\right) & \to & q + 9 & \to & q \\ \\ \end{array}
b
\begin{array}{ccccc} & ⬚ & & ⬚ & \\ 5 \left(r - 7\right) & \to & r-7 & \to & r \\ \\ \end{array}
c
\begin{array}{ccccc} & ⬚ & & ⬚ & \\ \dfrac{n + 6}{9} & \to & n+6 & \to & n \\ \\ \end{array}
d
\begin{array}{ccccc} & ⬚ & & ⬚ & \\ \dfrac{j}{2}-7 & \to & \dfrac{j}{2} & \to & j \\ \\ \end{array}
6

Consider the equation 4 x + 16 = 24.

a

Identify which operations should be done to make x the subject of the equation:\begin{array}{ccccc} &\text{Operation }1&&\text{Operation }2&\\ 4x + 16&\to&4x&\to&x \end{array}

b

Complete the following flow chart by applying the same operations to the right-hand side of the equation:\begin{array}{ccccc} &\text{Operation }1&&\text{Operation }2&\\ 24&\to&⬚&\to&⬚ \end{array}

c

Hence, find the value of x that will make the equation true.

7

Consider the equation 3 \left(p + 11\right) = 57.

a

Identify which operations should be done to make p the subject of the equation:\begin{array}{ccccc} &\text{Operation }1&&\text{Operation }2&\\ 3 \left(p + 11\right)&\to&p + 11&\to&p \end{array}

b

Complete the following flow chart by applying the same operations to the right-hand side of the equation:\begin{array}{ccccc} &\text{Operation }1&&\text{Operation }2&\\ 57&\to&⬚&\to&⬚ \end{array}

c

Hence, find the value of p that will make the equation true.

8

Consider the equation \dfrac{j}{7} - 3 = 7.

a

Identify which operations should be done to make j the subject of the equation:\begin{array}{ccccc} &\text{Operation }1&&\text{Operation }2&\\ \dfrac{j}{7} - 3&\to&\dfrac{j}{7}&\to&j \end{array}

b

Complete the following flow chart by applying the same operations to the right-hand side of the equation:\begin{array}{ccccc} &\text{Operation }1&&\text{Operation }2&\\ 7&\to&⬚&\to&⬚ \end{array}

c

Hence, find the value of j that will make the equation true.

9
For each of the following equations:
i

Describe the operations required to make the pronumeral the subject.

ii

Hence, solve the equation.

a
7 p + 6 = 41
b
2 x - 5 = 11
c
5 \left(n - 15\right) = 35
d
10 \left(p + 10\right) = 120
e
\dfrac{y}{8} + 3 = 5
f
\dfrac{x}{3} - 11 = 9
g
\dfrac{u + 22}{6} = 4
h
\dfrac{y - 10}{2} = 5
10

Explain how to solve the equation -2x - 4 = -18.

Two-step equations
11

Solve the following equations, remembering to check your answer:

a
2 w + 9 = 37
b

7 x + 17 = 45

c
2 x + 13 = 19
d

4 x + 9 = 25

e

2 t + 13 = 35

f

7 m + 18 = 39

g

12 p + 3 = 51

h

2x + 14 = 20

i

6y + 4 = 34

j

3q + 9 = 12

12

Solve the following equations:

a
6 m - 60 = 12
b
8q - 42 = 6
c

13 x - 2 = 24

d

5 t - 15 = 15

e

29 m - 8 = 21

f

11 x - 22 = 22

g

2p - 17 = 3

h

9q - 4 = 23

i

6n - 29 = 7

j

12m + 7 = 19

Equations with fractions and brackets
13

Solve the following equations:

a
\dfrac{x}{5} + 3 = 11
b
\dfrac{j}{8} - 17 = 3
c
\dfrac{k}{8} - 10 = 1
d
\dfrac{r}{6} - 3 = 4
e

\dfrac{x}{3} + 2 = 7

f

\dfrac{q}{2} + 18 = 25

g

\dfrac{s}{7} + 4 = 5

h

\dfrac{m}{5} + 13 = 22

i

\dfrac{p}{11} - 3 = 3

j

\dfrac{n}{8} - 5 = 4

k

\dfrac{p}{13} - 3 = 7

l

\dfrac{x}{12} + 2 = 6

m

\dfrac{s}{4} + 15 = 31

n

\dfrac{x}{6} + 7 = 17

o
\dfrac{t}{2} + 6 = 8
p
\dfrac{m}{2} + 11 = 16
14

Solve the following equations:

a
\dfrac{u + 7}{2} = 5
b
\dfrac{q - 10}{4} = 6
c
\dfrac{m + 11}{2} = 10
d

\dfrac{40 + m}{15} = 3

e

\dfrac{r+11}{5} = 7

f

\dfrac{m+26}{17} = 3

g

\dfrac{t-5}{4} = 8

h

\dfrac{q-17}{3} = 4

15

Solve the following equations:

a
5 \left(t + 9\right) = 60
b
4 \left(x + 5\right) = 20
c
2 \left(p + 14\right) = 58
d
3 \left(p + 9\right) = 48
e

9 \left( x + 2 \right) = 27

f

2 \left( p + 9 \right) = 28

g

6 \left( x + 5 \right) = 54

h
3\left(n + 5\right) = 24
i
4 \left(k - 4\right) = 48
j
3 \left(s - 16\right) = 21
k
9 \left(p - 3\right) = 0
l
2 \left(l - 12\right) = 10
m
4 \left(s - 29\right) = 4
n
4 \left(s - 8\right) = 16
o
3\left(s - 9\right) = 3
p

13 \left( s - 7 \right) = 143

Mixed equations
16

Solve the following equations:

a
\dfrac{8 t}{3} = 16
b
48 = 7z+13
c
\dfrac{3 y}{8} = 6
d

11 - \dfrac{x}{6} = 9

e

14 = 3 r - 7

f
48 = 6 \left(p+8\right)
g

3 \left( 12 + x \right) = 90

h

\dfrac{n-2}{6} = 10

i
6 = \dfrac{t}{5} - 3
j

8 s -17= 7

k
4 = 4 \left(s - 29\right)
l

8\left( 2 + s \right) = 32

m

10 = 2x + 6

n

5x-10= 15

o
\dfrac{x}{2}+7 = 9
p

11 = \dfrac{x+1}{2}

17

Explain how to solve the equation -12 = 8 + \dfrac{x}{5}.

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Outcomes

7.C2.3

Solve equations that involve multiple terms, whole numbers, and decimal numbers in various contexts, and verify solutions.

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