Solving Simultaneous Equations (Substitution Method)

In this lesson, you’ll learn how to solve simultaneous equations using substitution, where two equations are solved together to find the values of x and y.

What are simultaneous equations?

Simultaneous equations are when two equations are solved together to find the values that work in both equations at the same time.

In this lesson, we’ll focus on solving simultaneous equations using the substitution method, which is often the clearest place to start at GCSE.

Example Question:

We will work through this pair step by step:

Solve the simultaneous equations

$x + y = 5$
$2x + y = 8$

Exam Tip

Always rearrange the equation that is easiest to work with first.

Look for:

A variable with coefficient 1
An equation that is already nearly rearranged

This reduces the chance of mistakes.

Method – Solving Simultaneous Equations by Substitution

The substitution method solves simultaneous equations by replacing one variable with an expression from the other equation.

Follow these steps:

Rearrange one equation
Choose the equation that is easiest to rearrange and make one variable the subject. You are isolating one variable so it can be substituted into the other equation.
Substitute into the second equation
Replace that variable in the second equation with the expression you found. This creates a new equation with only one unknown.
Solve the new equation
Simplify, expand brackets if needed, and solve to find the value of the remaining variable.
Substitute back
Substitute this value into one of the original equations to find the other variable.
Check your solution
Substitute both values into the original equations to confirm they work. This step is strongly recommended in exams.

Worked Example

Solve:
$x+y=5$
$2x+y=8$

Step 1: Rearrange the first equation:

We’ll rearrange the simpler equation to make $x$ the subject.
$x+y=5$
$x=5-y$
(Subtract $y$ from both sides)

Step 2: Substitute into the second equation:

Now substitute $x=5-y$ into the second equation:

$2x + y = 8$
$2(5-y)+y=8$
(Replace $x$ with $5-y$ .)

Step 3: Expand and simplify

Expand the bracket:
$10 – 2y + y =8$

Combine like terms:
$10-y=8$
( $-2y+y=-y)$

Step 4: Solve for $y$

$10-y=8$
Subtract 10 from both sides:
$-y=-2$
Multiply both sides by $-1$
$y=2$
(Be careful with negative signs here.)

Step 5: Substitute back to find $x$

Use the rearranged equation:
$x=5-y$
Substitute $y=2$ :
$x=5-2$
$x=3$

Final answer:

The solution to the simultaneous equations is:
$x=3$
$y=2$

Always give both values – not just one.

Check your solution (important habit)

Substitute the values into both original equations.

First equation:
$3 + 2 = 5$ ✓

Second equation:
$2(3) + 2 = 8$
$6+2=8$ ✓

Both equations are satisfied, so the solution is correct.

Think Like a Mathematician

Rearranging equations can feel unfamiliar at first, but it follows clear algebra rules.

You are simply isolating one variable and reducing the problem step by step until only one unknown remains.

The order of the equations does not matter — logical steps do.

Common mistakes to avoid

Forgetting to substitute into the entire equation
Missing brackets when substituting
Errors when expanding brackets
Finding one variable but forgetting to substitute back
Sign errors when simplifying

Most mistakes occur during substitution and expansion — slow down at these steps.

Try one yourself

Solve the following pair:

$x + y = 7$
$x – y = 1$

Hint: Rearrange either equation first – both are equally easy this time.

Frequently Asked Questions

What is the substitution method?

The substitution method involves rearranging one equation and substituting it into the other to form a single equation with one variable.

When should I use substitution instead of elimination?

Substitution is usually easier when one equation already has a variable with coefficient 1 or is easy to rearrange.

Next Lesson

Continue → Solving Simultaneous Equations by Elimination