Solving Simultaneous Equations by Substitution

June 5, 2022

Differentiated Learning Objectives

All students should be able to solve a pair of equations simultaneously using the method of substitution.
Most students should be able to change the subject of a formula to solve a pair of equations simultaneously using the method of substitution.
Some students should set up a pair of equations and use the substitution method to solve real-life problems.

Links to Lesson Resources (Members Only)

Starter/Introduction

Students recap changing the subject of a formula. Please encourage students to work on mini-whiteboards to see their work. Changing the subject is necessary later in the lesson as students make x or y the subject of the equation to substitute it into the other equation.

Prompts / Questions to consider

How do we use the balance method to cancel out a multiply, divide, addition or subtraction?
When a term is the subject of an equation or formula, what does it mean?

Solving Simultaneous Equations by Substitution

Solving simultaneous equations by substitution is an alternative approach to the elimination method. The best method to use depends on the coefficients in the equations. We use the substitution method when x or y appears with a coefficient of one in either of the two equations.

I begin the central part of the lesson by discussing how we use the substitution method to go from two equations with two unknowns to a single equation with one unknown.

The demonstration questions on slide 2 are scaffolded, so at the start, equations are written in terms of x or y, progressing to later questions when students need to make x or y the subject of the equation before the substitution.

I work through questions a) and b) on the second slide as shown in the video above, then ask students to attempt questions c) in pairs on a single whiteboard. After looking at their boards, I feedback the class on any mistakes and then ask them to try question d). I encourage students to check their solutions by substituting their values for x and y back into one of the equations.

After working through the questions on slide 2, students work independently through the questions on slide 3 shown below.

Prompts / Questions to consider

Which unknown do we substitute into the other equation?
Does the equation need to be rearranged before it can be substituted?
How can we check our answers?

Plenary

The plenary challenges students to form their pair of equations by equating the two equal sides of a rectangle. When I present this question to the class, I give them about 3 minutes to develop an approach. If students struggle to get started, I demonstrate how to form an equation by equating the two equal widths. For example, 2x + y + 1 = x + 3y – 7, therefore x – 2y = 8. At this point, most of the students can work through the problem independently.

Prompts / Questions to consider

Which pairs of sides are equal?
How can we set up two equations by equating the equal sides?
With the equations set up, can the substitution method be used to solve them?

Differentiation

More able students could quickly progress to the later questions in the included worksheet, setting up equations from real-life problems. Less able students could focus on solving equations in which one equation is given x or y.