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How to Solve Simultaneous Equations Using the Substitution Method (Mathematics Lesson)

How to Solve Simultaneous Equations Using the Substitution Method

The substitution method is used to solve simultaneous equations.

It involves rearranging one of the simultaneous equations to make an unknown (x or y) the subject of the equation. This expression then replaces (or substitutes for) that unknown in the other simultaneous equation.

This leaves one equation with one unknown, which can be solved to find the unknown, and so find the other.

General Approach to Using the Substitution Method:

Step 1
Rearrange an equation to make x or y the subject.

Step 2
Substitute this expression in place of the unknown in the other equation.

Step 3
Solve the equation to find one unknown.

Step 4
Substitute the value for this unknown in either equation and solve to find the other unknown.

A Real Example of Solving Simultaneous Equations Using the Substitution Method

Question: What values of x and y solve the simultaneous equations below?



Step 1
Rearrange an equation to make x or y the subject.

Rearrange Equation (1) to make y the subject.

This is an algebraic equation that contains addition.

To make y the subject, look at what is being done to the y:



The opposite of adding x is subtracting x, and this should be done to both sides:



y is now the subject of the equation.

Step 2
Substitute this expression in place of the unknown in the other equation.

Having rearranged Equation (1) to find an expression for y, this expression can now be substituted for y in Equation (2):



Step 3
Solve the equation to find one unknown.

This is one equation with one unknown. It can be solved to find x.

First expand out the bracket:



Then collect like terms.

Collect the x's on the left hand side of the equation:



Collect the number terms on the right hand side. To do this, cancel the -5 on the left hand side by adding 5 to both sides:



This is an algebraic equation that contains multiplication.

The x is multiplied by 2, so divide both sides by 2 to find x:



Step 4
Substitute the value for this unknown in either equation and solve to find the other unknown. Substitute x = 3 into Equation (1):



This is an algebraic equation that contains addition.

To make y the subject, look at what is being done to the y:



Subtract 3 from both sides to solve y:



Both unknowns are now known.

The solution to the simultaneous equations is:



The slider below shows an example of solving simultaneous equations using the substitution method:
Interactive Test
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Note
WHAT IF THE EQUATIONS HAVE DIFFERENT COEFFICIENTS?

The substitution method works regardless of the of the unknowns in the simultaneous equations.

For example, to solve the simultaneous equations below:



For Step 1, one of these equations must be rearranged to make an unknown a subject.

Rearrange Equation (1) to make y the subject:

  • Subtract 4x from both sides, to leave 2y on its own on the left hand side of the equation:



  • Divide both sides by 2, the of the y:

For Step 2, substitute this expression for y into Equation (2):



This can then be solved to find x, and hence y.

CAREFUL WHEN EXPANDING BRACKETS

When substituting an expression for one of the unknowns into the other equation (Step 2) - it is always wise to place this expression in brackets.

Consider the Note above. By substituting for y into the equation, the following equations was obtained:



The next step is to expand the bracket (highlighted above in blue).

Each term in the bracket must be multiplied by the -3 in front of the bracket.

Multiply -3 by 4:



Then multiply -3 by -2x (paying careful attention to the signs):



The bracket has now been expanded: