How to calculate simultaneous equations: What should your students know?

Simultaneous equations are a fundamental part of algebra at Key Stage 3, so students will need to be fully prepared for what they may encounter in their GCSEs – and beyond, should they progress to A-level – both with and without a calculator. This should include both linear simultaneous equations and quadratic simultaneous equations, as well as the best methods for solving each.

Methods for solving simultaneous equations

Method 1: Substitution Method

For many students, the substitution method is the simplest option to understand, so may be useful as their first introduction to the topic. This involves finding the value of one variable in terms of the other variable from one equation. Students then substitute  this value in the second equation.

Example:

To use the substitution method, one of the variables must be able to be isolated in at least one of the equations. The substitution method is especially useful when equations are either given in the form, or can be simplified into the form:

x = ay + b

y = mx + n

Method 2: Elimination Method

The main alternative to substitution when solving simultaneous equations algebraically is the elimination method, which is one of the most common ways of tackling linear simultaneous equations. This is useful when the coefficients of none of the variables are the same and allows students to solve the equations without the need to isolate any of the variables.

It works by multiplying or dividing either one or both equations to make the coefficient of either the x-variable or y-variable the same in both equations, before using addition or subtraction to eliminate the variable whose coefficient is the same.

Example:

Method 3: Graphical Method

The third way to answer these questions is by displaying the equations as a graph. This offers a clear visual solution that many students find easier to understand. Solving equations graphically is made much simpler with the help of a powerful calculator such as the Casio fx-CG50.

A correctly plotted graph will show students a clear intersection point between the two equations, which can be used to easily identify the values on the x and y axes.

Examples of solving simultaneous equations with a calculator

For many students, the graphical method of solving equations may be the most intuitive and useful when access to a calculator is permitted. However, for students who do not have access to these tools, there are other options available.

To solve two linear simultaneous equations on the fx-CG50 graphic calculator, for example, follow these steps:

• Enter the Equation app
• Select Simultaneous and then the number of unknowns (2)
• Carefully enter each equation’s coefficients of x and y and the constants, allowing for rearrangement
• Press SOLVE to display the tabulated decimal values of x and y

Simultaneous equations can also be solved on a scientific calculator without using the graphical method. Here’s how to do so on the fx-991CW.

• Press the MENU button and scroll down to select the Equation function
• Select Simultaneous Equations by pressing 1
• Choose the number of unknowns
• Use the Coefficient Editor to input the values of the coefficients,
• Press EXE once to see the solution for the first unknown, and again for the second

Common mistakes when solving simultaneous equations

When following the process correctly, students should be able to get to grips with solving simultaneous equations. However, there are a few common mis-steps that can happen along the way that lead them to an incorrect answer. Knowing what these are and why students make such errors can help you assist your students with the topic.

Some common mistakes to look out for include:

• Incorrectly eliminating variables: Using the wrong operations to eliminate a variable – for example, a student adding when they should subtract or vice-versa.
• Mishandling negative numbers: It can be easy for students to make mistakes when adding, subtracting, multiplying or dividing negative numbers, and even a small error can result in the wrong answer. To avoid this, it can be helpful to work out these steps separately.
• Not multiplying every term: Forgetting to multiply each term within the equation by the same number.

Using their calculator effectively can help students eliminate many of these errors, so it’s important they know how to operate their devices properly when solving simultaneous equations.

Real-world applications of simultaneous equations

As with many aspects of maths, it’s often helpful to provide some context when presenting algebraic simultaneous equations. Being able to lift the questions off the page and show how the theory can be applied to real-world applications can help learners engage with the materials and spark inspiration.

Among the situations you could highlight for students when devising questions for simultaneous equations are:

Route planning: Working out the most effective route for a journey by taking into account distances and average speeds for different parts of the route can produce various insights, such as faster and more efficient options.

Finding the best deal: Inputting options such as interest rates and monthly payments as the variables in simultaneous equations can help answer questions such as what loans, investments or phone plans offer the best overall deal.

Understanding supply and demand: In economics, simultaneous equations have a range of uses, including analysing the relationships between price and quantity of products to determine supply and demand and understand how changes to one variable will affect the overall market.