Solving two-step equations with positive unknowns

Home learning focus

In this lesson, you will learn how to solve two-step equations where the solution is positive.

This lesson includes:

• two activities

Created in partnership with NCETM

Learn

Two-step equations

Equations where the unknown has been operated on twice are called two step equations. For example:

3x – 7 = 20

To solve the equation, you need to figure out what operations (add, subtract, multiply, divide) have been performed and in what order, and then undo them to find the value of the unknown.

In this example, x has been multiplied by 3, and then 7 has been subtracted. So, the two operations are multiply by 3 and subtract 7.

To help work this out, you can use a balance and a function machine.

Balance

Let's use the two-step equation 2x + 3 = 11

You can think of this like a balance with 2x + 3 on the left-hand side and 11 on the right-hand side.

You could imagine this as two identical unknown weights (2x) and three 1kg weights on the left, and eleven 1kg weights on the right. For example:

1. 2x + 3 = 11

You can perform the same operation to both sides to maintain the balance and keep doing this until the equation reads x = ?.

2. Removing three 1kg weights from both sides gives:

3. Halving the weight on each side gives:

So, in the two-step equation 2x + 3 = 11, x = 4.

Function machines

You can also use a function machine to work out the value of x in two-step equations.

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

The equation 2x + 3 = 11 can be shown on a function machine by writing out the functions that have been applied to x in the order they happened.

2x + 3 = 11 means x has been multiplied by 2 and then 3 has been added. The answer is 11.

Using the function machine below, you can decide which order the operations have been applied to x and then apply the reverse of each of them in the reverse order to find the value of x.

So, in the two-step equation 2x + 3 = 11, x = 4.

Practise

Activity 1

x = 5

Copy and complete the spider diagram with x = 5 in the middle to create other equations where the solution is x = 5.

Check that x = 5 really is the solution to all of the equations you have written.

Can you explain why?

Activity 2

Worksheet: Solving equations

Complete this worksheet provided by Beyond about solving equations involving positive unknowns.