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What are equations?

Equations are made up of two expressions on either side of an equals sign, such as:

\(x + 1 = 2\)

To solve an equation, you aim to find the value of the missing number.

Example

'I think of a number, add four, and the answer is seven.'

Written algebraically, this statement is \(x + 4 = 7\), where \(x\) represents the number you thought of.

\(x + 4 = 7\) is an example of an algebraic equation where \(x\) represents the unknown number.

The number you first thought of must be three (\(3 + 4 = 7\)).

Therefore, \(x = 3\) is the solution to the equation \(x + 4 = 7\).

Example 1

Suppose you are trying to find out how many sweets are in the bag shown here.

Each of the sweets weighs the same amount.

Let \({x}\) equal the amount of sweets in the bag.

We can ignore the weight of the bag.

By subtracting three sweets from each side, the scales remain balanced.

You can now see that one bag is equivalent to two sweets. Written algebraically, this is:

\(x + 3 = 5\)

Subtract \(3\) from both sides, to give:

\(x = 2\)

There are 2 sweets in the bag.

To find the equivalent of one bag, divide both sides by two.

Written algebraically, this is:

\(2x = 6\)

Divide both sides by \(2\) to give:

\(x = 3\)

Multiples of an unknown

Sometimes an equation will have multiples of an unknown, eg \(5y = 20\).

To solve this you need to get the unknown on its own. To do this, divide both sides by \({5}\).

\(5y = 20\)

\(5y \div 5 = 20 \div 5\)

\(y = 4\)

Check your answers

When solving algebraic equations, always check your answers.

For example, if you think that the answer to the equation \(x + 5 = 12\) is \(x = 7\), then check it by replacing \(x\) with \(7\).

\(7 + 5\) does equal \(12\), so your answer is correct.

Sometimes an equation will have multiples of an unknown and other numbers, eg \(3x + 2 = 8\).

In equations of this type, your aim is to get all the \({x}\)s (or unknowns) on one side and all the numbers on the other.

Example

Let's solve the equation:

\(3x + 2 = 8\).

We can represent this in the diagram opposite, where each bag represents the unknown value \( x \).

Therefore:

\(3x + 2 = 8\)

We want to get the \(x\) on its own.

Start by subtracting \(2\) from both sides.

\(3x + 2 - 2 = 8 - 2\)

\(3x = 6\)

Then divide by \(3\) on both sides:

\(x = 2\)

Example

Solve the equation:

\(2x + 2 = x + 4\)

The equation \(2x + 2 = x + 4\) is represented by the diagram opposite.

As before, the bags represent the unknown value (\(x\)) and the sweets represent the numbers in the equation.

Aim to get the unknown value on just one side of the equation, so begin by subtracting \(x\) (taking one bag away) from each side.

Now you have the type of equation that you recognise, so all you need to do is subtract \(2\) from both sides.

Written algebraically, this becomes:

\(2x + 2 = x + 4\)

Subtract \(x\) from both sides to give:

\(x + 2 = 4\)

Subtract \(2\) from both sides to give:

\(x = 2\)

Key Point

If the term in \(x\) is negative, tackle the equation in the same way - aim to get all \(x\)s on one side of the equation.

Example 1

Solve the equation:

\(5x - 2 = 12 - 2x\)

Your aim is to get all the unknown \(x\) terms on one side of the equation, so start by adding \(2x\) to both sides:

\(7x - 2 = 12\)

Next add \(2\) to both sides:

\(7x = 14\)

And finally, divide by \(7\) to give:

\(x = 2\)

Again, you can check your answer in the original equation.

So substitute \(x = 2\) back into:

\(5x - 2 = 12 - 2x\)

\((5 \times 2) - 2 = 12 - (2 \times 2)\)

\(10 - 2 = 12 - 4\)

\(8 = 8\)

This makes sense, so the value \(x = 2\) is correct.

Example 2

Solve the equation:

\(4x = 10 - x\)

Add \(x\) to both sides to give:

\(5x = 10\)

To find \(x\), divide both sides by \(5\):

\(x = 2\)

Example 1

Expand the bracket:

\(2(3a + 5)\)

Multiply everything inside the bracket by the number outside to give:

\((2\times3a)+(2\times5)\)

Simplify this to:

\(6a + 10\)

Example 2

Solve the equation: \(2(b-2) = -3(2b-4)\)

First expand the brackets: \(2b-4= -6b+12\)

Then add \(6b\) to both sides: \(8b-4=12\)

Add 4 to both sides: \(8b=16\)

Finally divide both sides by \(8\): \(b=2\)

To solve equations written as fractions, you must do the same thing to each side of the equation to get the answer.

It is useful to know that \(\frac{1}{2}x\) is the same as \(\frac{x}{2}\), and \(\frac{1}{4}x\) is the same as \(\frac{x}{4}\), etc.

For example look at the equation:

\(\frac{1}{2}x = 3\)

It can be rewritten as:

\(\frac{x}{2} = 3\)

The number at the bottom is \(2\), so multiply both sides by \(2\).

\(x = 6\)

If an equation has fractions on both sides, multiply both sides by the lowest common multiple (LCM) of the denominators…

Example

Solve this equation:

\(\frac {(x - 3)} {2} = \frac {(x - 2)} {3}\)

Solution

One side is divided by \(2\).

The other side is divided by \(3\).

The lowest common multiple of \(2\) and \(3\) is \(6\), so multiply both sides by \(6\).

\(\frac {(x - 3)} {2} \times 6 = \frac {(x - 2)} {3} \times 6\)

\(3(x - 3) = 2(x - 2)\)

Expand the brackets:

\(3x - 9 = 2x - 4\)

Take \(2x\) from both sides:

\(x - 9 = - 4\)

Add \(9\) to both sides:

\(x = 5\)