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Solving Simultaneous Equations Using Substitution

(KS4, Year 10)

Simultaneous equations are a set of several equations with several unknowns. We can use the substitution method to find the values of the unknowns which solve both equations at the same time. An unknown from one equation is substituted into the other equation, allowing one unknown to be found in one equation.

solve simultaneous equations substitution

How to Solve Simultaneous Equations Using the Substitution Method

Solving simultaneous equations using the substitution method is easy.

Question

Solve the simultaneous equations shown below using the substitution method.
solve simultaneous equations substitution example 1 We can substitute Equation (1) into Equation (2).

Step-by-Step:

1

Equation (1) tells us that y = 2x. Substitute this value of y into Equation (2). solve simultaneous equations substitution example 1 step 1
x + 2x = 3

2

Solve for x.
x + 2x = 3
3x = 3 Add the like x terms
3x ÷ 3 = 3 ÷ 3 Divide both sides by 3
x = 1
x = 1 is a solution to the simultaneous equations.

3

Substitute the variable we have just found (x = 1) into one of the equations. Solve for y.
y = 2x Substitute into Equation (1)
y = 2( 1 )
y = 2 × 1
y = 2
y = 2 is a solution to the simultaneous equations.

Answer:

We have solved the simultaneous equations:

x = 1, y = 2 solves

y = 2x

x + y = 3

A Real Example of How to Solve Simultaneous Equations Using the Substitution Method

The example above was simple. Equation (1) told us what "y = ". In the following example, we will have to find what "y = " by rearranging one of the equations. Note: you could find what "x = "... it doesn't matter which unknown you substitute.

Question

Solve the simultaneous equations shown below using the substitution method. solve simultaneous equations substitution example 2

Step-by-Step:

1

Rearrange Equation (2) to find "y =".
x − y = 1
x − y + y = 1 + y Add y to both sides
x = 1 + y
x − 1 = 1 + y − 1 Subtract 1 from both sides
x − 1 = y
y = x − 1
We have rearranged x − y = 1 to find what "y = ". y = x − 1

2

Rearranged Equation (2) tells us that y = x − 1. Substitute this value of y into Equation (1).

solve simultaneous equations substitution example 2 step 2
x + ( x − 1 ) = 5

3

Solve for x.
x + x − 1 = 5
2x − 1 = 5 Add the like x terms
2x − 1 + 1 = 5 + 1 Add 1 to both sides
2x = 6
2x ÷ 2 = 6 ÷ 2 Divide both sides by 2
x = 3
x = 3 is a solution to the simultaneous equations.

4

Substitute the variable we have just found (x = 3) into one of the equations. Solve for y.
x + y = 5 Substitute into Equation (1)
3 + y = 5
3 + y − 3 = 5 − 3 Subtract 3 from both sides
y = 2
y = 2 is a solution to the simultaneous equations.

Answer:

We have solved the simultaneous equations:

x = 3, y = 2 solves

x + y = 5

x − y = 1

Lesson Slides

The slider below shows another real example of how to solve simultaneous equations using the substitution method.
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This page was written by Stephen Clarke.

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