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We need to find a value for \(a\) which keeps this balance.

We could try different values until we hit on the correct value to balance the equation, but it is much better to have a method.

To solve a linear equation such as \(3a + 8 = 26\)

• Change the equation, keeping it balanced, so that all the unknown terms are on one side and all the numbers are on the other side
• Divide or multiply to find the value of the unknown

Check the answer by using your value in the original equation - it should be balanced. It's good to check!

To solve \(3a + 8 = 26\)

1. Subtract 8 from each side of the equation. (Subtracting the same value from each side keeps the equation balanced.)

\(3a + 8 - 8 = 26 - 8\)

which gives \(3a = 18\)

This leaves just \(3a\) on the left hand side.

2. Find the value of \(a\) by dividing by 3 remember that \(3a\) means \(3 \times a\)

\(a = 18 \div 3\)

\(a = 6\)

Check the answer

Look at the original equation \(3a + 8 = 26\)

Find the value of \(3a + 8\) when \(a = 6\)

This should be 26

When \(a = 6\)

\(3a = 3 \times 6\)

\(= 18\)

\(3a + 8 = 18 + 8\)

\(= 26\)

Both sides are equal to \(26\), so the answer is correct.

Example

Solve the equation \(4c - 7 = 5\)

Solution

We want just \(4c\) on the left-hand side,

so add 7 to each side

\(4c - 7 +7 = 5 + 7\)

\(4c = 12\)

Divide by 4

\(c = 12 \div 4\)

\(c = 3\)

Check answer

Look at the original equation: \(4c - 7 = 5\)

When \(c = 3\)

\(4c = 4 \times 3\)

\(= 12\)

and \(4c - 7 = 12 = 7\)

\(= 5\)

Both sides are equal to \(5\), so the answer is correct

Example

Solve the equation \(\frac{q}{3} + 11 = 18\)

Solution

We want just \(\frac{q}{3}\) on the left hand side,

so subtract 11 from each side.

\(\frac{q}{3} + 11 - 11 + 18 - 11\)

\(\frac{q}{3} = 7\)

Multiply by 3

\(q = 3 x 7\)

\(q = 21\)

Check answer

Look at the original equation\(\frac{q}{3} + 11 = 18\)

When \(q = 21\),

\(\frac{q}{3} = 21 \div 3\)

\(= 7\)

\(\frac{q}{3} + 11 = 7 + 11\)

\(= 18\)

Both sides are equal to 18 so the answer is correct.

Example

The diagram shows an isosceles triangle with the two equal sides, each \(d\) centimetres long.

The third side is 12cm long.

Given that the perimeter of the triangle is 27cm, form an equation to find the value of \(d\).

Solution

First set up an equation.

The perimeter is the distance right around the triangle, so add all the sides together.

\(Perimeter = d + d + 12\)

This simplifies to \(2d + 12\)

We were told that the perimeter is 27cm so we can write the equation:

\(2d + 12 = 27\)

To solve

subtract 12 from each side of the equation

\(2d + 12 - 12 = 27 - 12\)

which gives \( 2d = 15\)

Divide by 2 to find the value of \(d\)

\(d = 15 \div 2\)

\(= 7.5\)

Check answer

\(d\) = 7.5cm

Go back to the equation for the perimeter

\(2d + 12 = 27\)

When \(d\) = 7.5

\(2d = 2 \times 7.5\)

\(= 15\)

\(2d + 12 = 15 + 12\)

\(= 27\)

Both sides are equal to 27, so the answer is correct.

Example

Solve the equation \(3(5p - 2) - 4 = 35\)

Solution

In this example, the left hand side must be simplified before carrying out the steps to solve the equation.

First multiply out the bracket \(3(5p - 2) = 15p - 6\)

Put this into the equation \(15p - 6 - 4 = 35\)

Simplify the left hand side \(15p - 10 = 35\)

Solve the equation - add 10 to both sides

\(15p - 10 + 10 = 35 + 10\)

\(15p = 45\)

Divide by 15

\(p = 45 \div 15\)

\(p = 3\)

Check answer

Look at the original equation: \(3(5p - 2) - 4 = 35\)

When \(p = 3\),

\(3(5p - 2) - 4 = 3(15 - 2) - 4\)

\(= 3 \times 13 - 4\)

\(= 39 - 4\)

\(= 35\)

Both sides are equal to \(35\) so the answer is correct.

Your answer for the last example could be as short as this:

\(3(5p - 2) - 4 =35\)

\(15p - 6 - 4 = 35\)

\(15p - 10 = 35\)

\(15p = 45\)

\(p = 3\)

Check:

\(3(15 - 2) - 4 = 39 - 4\)

\(= 35\)