Text
Unit F2 Section 3
Solving Linear Equations

Most equations require a number of steps to solve them. These steps must be logical so that the new equation still balances. Whatever you do to one side of an equation you must do the same to the other side. The following examples illustrate these steps.

Worked Examples

1

Solve the following equations.

(a)

3x + 7 = 13

First subtract 7 from both sides of the equation.

3x + 7 = 13
3x + 7 − 7 = 13 − 7
3x = 6

Next divide both sides of the equation by 3.

=
x = 2
(b)

5x − 8 = 13

First add 8 to both sides of the equation.

5x − 8 = 13
5x − 8 + 8 = 13 + 8
5x = 21.

Then divide both sides of the equation by 5.

=
x =
=
(c)

− 2 = 3

First add 2 to both sides of the equation.

− 2 = 3
− 2 + 2 = 3 + 2
= 5

Then multiply both sides of the equation by 5.

× 51 = 5 × 5
x = 25
(d)

4(x − 3) = 8

First remove the brackets, multiplying each term inside the bracket by 4.

4(x − 3) = 8
4x − 12 = 8

Then add 12 to both sides of the equation.

4x − 12 + 12 = 8 + 12
4x = 20

Finally divide both sides by 4.

=
x = 5

Sometimes equations may contain the letter x on both sides of the equation or a − x term. The following examples show how to deal with these cases.

2

Solve these equations.

(a)

4x + 6 = 3x + 10

As x appears on both sides of the equation, first subtract 3x from both sides.

4x + 6 = 3x + 10
4x + 6 − 3x = 3x + 10 − 3x
x + 6 = 10

Then subtract 6 from both sides.

x + 6 − 6 = 10 − 6
x = 4
(b)

6 − 2x = 8

As the left-hand side contains −2x , add 2x to both sides.

6 − 2x = 8
6 − 2x + 2x = 8 + 2x
6 = 8 + 2x

Then subtract 8 from both sides.

6 − 8 = 8 + 2x − 8
−2 = 2x

Finally divide both sides by 2.

=
−1 = x   or   x = −1
(c)

4x − 2 = 8 − 6x

As one side contains −6x, add 6x to both sides.

4x − 2 = 8 − 6x
4x − 2 + 6x = 8 − 6x + 6x
10x − 2 = 8

Then add 2 to both sides of the equation.

10x − 2 + 2 = 8 + 2
10x = 10

Finally divide both sides by 10.

=
x = 1
3

Use the information in the diagram to write down an equation and then find the value of x.

The three angles shown must add up to 360°, so

170 + 2x + 50 + x − 10 = 360
210 + 3x = 360

Subtracting 210 from both sides gives

210 + 3x − 210 = 360 − 210
3x = 150

Then dividing both sides by 3 gives

=
x = 50
4

Arianne, Jovan and Kerry were playing a card game.

Arianne scored x points.

Jovan scored 3 points fewer than Arianne.

Kerry scored twice as many points as Jovan.

Together they scored 39 points.

(a)

Write down, in terms of x, an expression for the number of points scored by Kerry.

Number of points scored by Jovan = x − 3

Number of points scored by Kerry = 2(x − 3) = 2x − 6
(b)

Write an equation which may be used to find the value of x.

Total number of points = x + (x − 3) + 2(x − 3) = 39
= x + x − 3 + 2x − 6 = 39
= 4x − 9 = 39
(c)

How many points did Arianne score?

Solving,

4x − 9 = 39
4x = 39 + 9
4x = 48
x = 12

So Arianne scored 12 points.

Exercises

Solve each of these equations.

(a)
3x + 6 = 48 x =
(b)
5x − 6 = 39 x =
(c)
2x − 6 = 22 x =
(d)
6x − 7 = 41 x =
(e)
8x − 3 = 29 x =
(f)
6x + 12 = 20 x =
(g)
4x + 18 = 2 x =
(h)
5x + 10 = 5 x =
(i)
3x + 6 = 1 x =
(j)
5(x + 2) = 45 x =
(k)
3(x − 2) = 12 x =
(l)
2(x + 7) = 10 x =
(m)
3(2x − 1) = 57 x =
(n)
3(2x + 7) = 27 x =
(o)
5(5x + 1) = 20 x =
(p)
4(2x + 3) = −8 x =
(q)
5(3x − 1) = −2 x =
(r)
2(8x + 5) = −2 x =
(s)
6x − 8 = −26 x =
(t)
4(x + 15) = 60 x =
(u)
5x − 8 = −10 x =
(v)
− 1 = 8 x =
(w)
+ 2 = 7 x =
(x)
+ 1 = 3 x =

Solve these equations.

(a)
2x + 6 = x + 3 x =
(b)
4x − 8 = 5x − 2 x =
(c)
6x + 7 = 2x + 20 x =
(d)
x + 6 = 2x − 8 x =
(e)
3x + 7 = 2x + 11 x =
(f)
10x + 2 = 8x + 22 x =
(g)
6 − x = 5 x =
(h)
2 − x = 5 x =
(i)
3 − x = −10 x =
(j)
14 − 3x = 5 x =
(k)
10 − 2x = 2 x =
(l)
4 − 3x = 2 x =
(m)
x + 2 = 8 − x x =
(n)
x + 4 = 10 − 2x x =
(o)
x + 4 = 9 − 2x x =
(p)
8 − x = 12 − 2x x =
(q)
22 − 4x = 18 − 2x x =
(r)
3 − 6x = 2 − 4x x =
(s)
3(x + 2) = 5(x − 2) x =
(t)
4 = 8 − x =
(u)
3 − = −5 x =
(v)
4(x − 2) = 3(x + 2) x =
(w)
5 = 18 − x =
(x)
2 − = 1 − x =

You ask a friend to think of a number, double it and add 10. His answer is 42. If x is the number your friend thought of, write down the relevant equation and find x.

=

x =

Six teams enter a competition. There are x members in each team. If 8 people drop out and 34 complete the competition, write down an equation and solve it to find the number in each team at the start of the competition.

=

x =

Adam thinks of a number. He doubles it and then adds 5. The answer is 17. What was his number?

(a)

Write, in symbols, the rule,

To find y, double x and add 1.

=
(b)

Use your rule from part (a) to calculate the value of x when y = 9.

x =

Clifton uses this rule,

Start with a number, divide it by 2 and then add 3. Write down the result.

(a)

What is the result when Clifton starts with 8?

(b)

What number did Clifton start with when the result is 5?

Solve the equation

11x + 5 = x + 25

x =

The lengths of the sides of a triangle are

x cm, (x + 3) cm and (x − 2) cm.

(a)

What is the perimeter of the triangle in terms of x?

cm
(b)

The triangle has a perimeter of 22 cm.

(i)

Write down an equation in x.

=
(ii)

Use your equation to find the length of each side of the triangle.

cm,   cm   and   cm

A carton of orange juice costs x pence.

(a)

Write, in terms of x, the cost of two cartons of orange juice.

pence

The cost of a carton of pineapple juice is 20 pence more than the cost of a carton of orange juice.

(b)

Write in terms of x, the cost of a carton of pineapple juice.

pence

Sam pays £3.40 for three cartons of orange juice and two cartons of pineapple juice. He writes down the correct equation

3x + 2(x + 20) = 340

(c)

Solve this equation to find the cost of a carton of orange juice.

pence