# 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.

Click here for the answers.

# There's more to learn

Have a look at these other resources around the BBC and the web.