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Solving linear equations - AQANumber machines

Forming, using and solving equations are skills needed in many different situations. From balancing accounts to making sense of a mobile phone bill, solving equations is a vital skill.

Part of MathsAlgebra

Number machines

A number machine is a way of writing rules using a flow diagram.

The equation \(3j - 6 = 9\) can be shown on a number machine by writing out the functions that have been applied to \(j\) in the order they happened. BIDMAS means multiplications happen before subtractions.

\(3j - 6 = 9\) means \(j\) has been multiplied by 3 and then 6 has been subtracted. The answer is 9.

On a number machine, this would look like this.

Number machine flowchart showing j x 3 - 6 = 9.

Example

Create a number machine for the equation \(6j + 7 = 43\).

\(6j\) means \(6 \times j\), so \(j\) has been multiplied by 6. Then 7 has been added. The answer is 43.

Begin with the letter \(j\), and then show, in order, what functions have been applied to the letter.

Number machine flowchart showing j x 6 + 7 = 43.

Using number machines to solve equations

Number machines can be used to solve equations by reversing, or finding the of the steps.

Example

Solve the equation \(3j - 6 = 9\) using a number machine.

Firstly, write the equation as a number machine.

Number machine flowchart showing j x 3 - 6 = 9.

Then, find the inverse of each step. This means turning multiplications into divisions and subtractions into additions. Also, instead of starting at the beginning of the number machine, now start at the end.

Number machine flowchart showing j = / 3 + 6 9.

Now follow the number machine in reverse:

\(9 + 6 = 15\)

\(15 \div 3 = 5\)

So: \(j = 5\)

Check this answer by substituting \(j = 5\) into the original equation:

\(3j - 6 = 9\)

\(3 \times 5 - 6 = 9\)

\(15 - 6 = 9\)

\(9 = 9\)

The equation balances, so \(j = 5\) is the correct answer.

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