 1.02 Solving linear equations

Worksheet
One step equations
1

Solve for x:

a

x + 4 = 9

b

7 = x - 1

c

- 5 x = 30

d

\dfrac{x}{8} = 6

e

- 5 = \dfrac{x}{5}

f

4 x = - 28

g

\dfrac{1}{7}x = 6

h

x - 1.39 = 8.67

2

State whether the following statements are true or false:

a

x = 9 is a solution for the equation x - 10 = - 1

b

x = 0 is a solution for the equation x - 4 = - 5

c

x = 6 is a solution for the equation x + 1 = 7

d

x = 3 is a solution for the equation x - 3 = - 3

3

Find the value of t if 10 t = 60.

4

Solve - 6 = -\dfrac{y}{7}.

5

The following equations contain a geometric figure which represents a non-zero real number.

If \triangle =- x, state whether the following are true or false:

a

\triangle =x

b

x=-(-\triangle)

c

x=\triangle

d

x=-\triangle

Two step equations
6

State whether the following statements are true or false:

a

x = 6 is a solution for the equation 2 \left(x - 3\right) = - 8.

b

x = 8 is a solution for the equation 7 \left(x - 6\right) = 14.

c

x = 8 is a solution for the equation 3 \left(x - 6\right) = 6.

d

x = 8 is a solution for the equation 9 = 7 \left(x - 7\right).

7

Solve the following equations:

a

8 m + 9 = 65

b

5 x - 8 = 2

c

7 x + 14 = 0

d

- x - 7 = 7

e

- 6 x + 4 x - 3 x = 0

f

- 6 + 2 k = 10

g

5 \left(y + 1\right) = 25

h

3 \left(5 - x\right) = 0

i

\dfrac{x}{2} + 8 = 10

j

\dfrac{x}{2} - 3 = 2

k

\dfrac{x + 9}{7} = 4

l

\dfrac{c - 4}{4} = 7

m

- \dfrac{1}{8} x = 6

n

\dfrac{x}{6} = \dfrac{5}{3}

o

\dfrac{x}{6} = 2\dfrac{2}{3}

p

\dfrac{7.8}{7.5} = \dfrac{x}{5}

8

If the equation 5 y + 8 = c has a solution of y = 5, find the value of c.

Equations with three or more steps
9

Solve the following equations:

a
9 x - 45 = 4 x
b
3 \left(x + 6\right) = x
c
4 x + 24 = x - 9
d

5 x + 2 = 3 x + 22

e
6 x + 9 = 10 x - 3
f

8 x - 11 = 4 x

g

- 107 - 8 x = - 17 x + 19

h

- 64 - 9 x = - 24 - x

i
9 p - 6 = 3 p + 24
j
12 v + 6 = 110 - v
k
36 - 2 k = 6 k + 20
l
12 r + 8 = 116 - 6 r
m

2 \left(x - 2\right) = x - 4

n

4 \left(x + 9\right) = x - 9

o

3 x = - 13 - 4 \left( 2 x + 5\right)

p
3 \left(x + 1\right) + 2 = -1
q

- 2 \left(x + 2\right) - 6 = - 20

10

Solve the following equations:

a
x - 2 \left(x + 3\right) = -1
b

3 x + 2 \left( 3 x + 1\right) = 11

c

5 x - 2 \left( - 4 x - 5\right) = - 42

d
3 \left(x + \dfrac{3}{2}\right) + 3 = - \dfrac{15}{2}
e

7 \left(x - 6\right) = 5 \left(x + 2\right)

f

2 \left( 2 x + 5\right) = 3 \left(x + 5\right)

g

5.3 \left( 3 x + 55\right) = 4.9 \left(x + 55\right)

h

3 \left(x + 6\right) + 3 \left(x + 24\right) = 12

i

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

j

4 \left(x + 10\right) - 2 \left(x + 8\right) = 0

k

5 \left( 2 x + 2\right) = - 3 \left( 4 x - 5\right) + 5 x

l

9 \left(x - 9\right) - \left(x + 54\right) = 63 - \left(x - 63\right)

11

Consider the equation 3 k + 4 = A k - 2. If k = 6, find the value of A.

Equations with fractions
12

Solve \dfrac{2}{1.1} = \dfrac{5}{x}.

13

State whether we can immediately use cross multiplication to solve the following equations:

a

\dfrac{5 - x}{6} = \dfrac{2 + x}{7}

b

\dfrac{2}{3} - x = \dfrac{5 + x}{5}

14

Solve the following equations:

a

\dfrac{ x - 1}{3} = \dfrac{ x}{4}

b

\dfrac{ x}{4} = \dfrac{2x+1}{5}

c

\dfrac{2 x - 1}{3} = \dfrac{ x - 2}{4}

d

\dfrac{8 x - 2}{3} = \dfrac{6 x - 3}{4}

15

Solve the following equations:

a
\dfrac{11 x + 22}{8} = 33
b
\dfrac{3 x - 15}{5} = 9
c
- \dfrac{6 x}{5} + 5 = 35
d
- n - \dfrac{7}{8} = 3
e

\dfrac{5 x}{6} + 4 = 21.5

f
6 x - \dfrac{11}{7} = \dfrac{409}{7}
g
\dfrac{y - 4.3}{- 4} = 7.7
h
\dfrac{11.8 x + 31.86}{4} = - 106.2
16

Solve the following equations:

a

\dfrac{10 x - 26}{2} + 3 = 4 x

b

x + \dfrac{5 x-1}{4} = 1

c

\dfrac{n}{2} + \dfrac{n}{3} = 15

d

\dfrac{x}{5} - \dfrac{x}{2} = 3

e

\dfrac{9 x}{3} + \dfrac{9 x}{2} = - 5

f

\dfrac{5 x}{4} - 6 = \dfrac{3 x}{9}

g

\dfrac{3 x}{2} + 5 = \dfrac{2 x}{3}

h

\dfrac{2 x}{3} - 2 = \dfrac{5 x}{2} + 4

17

A construction company has spent \$22\,500\,000 to develop new cranes, and wants to limit the cost of development and production of a single crane to \$6000.

Given that the production cost of each crane is \$3000, the cost for development and production of x cranes is given by 3000 x + 22\,500\,000 dollars. a Write an expression for the cost of one crane. b If the cost of one crane must equal \$6000, find x, the number of cranes that must be sold.