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2 Linear equations
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Australia
Year 10

2.08 The substitution method

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Introduction

In order to solve for a variable in an equation, we need to be able to isolate that variable. If we are given a single equation with two unknown variables, it is impossible to solve for either value.

However, when we are given two equations, both with the same two variables, we are able to solve for both variables.

Substitute variables

When solving equations simultaneously, we are looking for some values for our variables which make both equations true. This means that in two equations in terms of x and y, each variable can be considered to be some fixed value for both equations- even if we don't know what that value is yet.

By assuming that the values for x and y in one equation are the same as for the other equation, we are able to substitute the value of a variable from one equation into the other.

Examples

Example 1

We want to solve the following system of equations using the substitution method.

\displaystyle y\displaystyle =\displaystyle x+11Equation 1
\displaystyle y\displaystyle =\displaystyle 3x+19Equation 2
a

First solve for x.

Worked Solution
Create a strategy

Substitute the expression for y from one equation into the other.

Apply the idea

We can substitute the expression that y is equal to in Equation 1, x+11, into the variable y in Equation 2 and solve for x:

\displaystyle x+11\displaystyle =\displaystyle 3x+19Substitute Equation 1 into Equation 2
\displaystyle 11\displaystyle =\displaystyle 2x+19Subtract x from both sides
\displaystyle -8\displaystyle =\displaystyle 2xSubtract 19 from both sides
\displaystyle x\displaystyle =\displaystyle -4Divide both sides by 2
b

Solve for y.

Worked Solution
Create a strategy

Substitute the value of x into one of the equations and solve for y.

Apply the idea
\displaystyle y\displaystyle =\displaystyle -4+11Substitute x=-4 into Equation 1
\displaystyle =\displaystyle 7Evaluate
Reflect and check

To double check that x=-4, y=7 is the solution to Equations 1 and 2 simultaneously we can substitute the values into both equations:

\displaystyle 7\displaystyle =\displaystyle -4+11Substitute x=-4, \, y=7 into Equation 1
\displaystyle 7\displaystyle =\displaystyle 7Evaluate the right side
\displaystyle 7\displaystyle =\displaystyle 3\times (-4)+19Substitute x=-4, \, y=7 into Equation 2
\displaystyle 7\displaystyle =\displaystyle 7Evaluate the right side

Since both equations hold true, we have confirmed that x=-4, y=7 is the simultaneous solution to the equations.

Idea summary

To solve simultaneous equations using the substitution method:

  • Substitute the expression for one variable, usually y, into the other equation.

  • Solve the new equation for the other variable, usually x.

  • Substitute the result back into one of the equations to find the value of the other variable.

Rearrange and substitute

In the example above, both equations already had y as the subject so it was easy to substitute it from one equation into the other. When this is not the case, we can choose one of the variables to isolate in one equation so that we can substitute it into the other.

Examples

Example 2

We want to solve the following system of equations using the substitution method.

\displaystyle -7p+2q \displaystyle =\displaystyle -\dfrac{13}{10}Equation 1
\displaystyle -21p+10q \displaystyle =\displaystyle -\dfrac{9}{10}Equation 2
a

Solve for q.

Worked Solution
Create a strategy

Isolate p in one of the equations, then substitute it into the other.

Apply the idea

We can start by isolating p in Equation 1.

\displaystyle -7p+2q\displaystyle =\displaystyle -\dfrac{13}{10}Write Equation 1
\displaystyle -7p\displaystyle =\displaystyle -\dfrac{13}{10}-2qSubtract 2q from both sides
\displaystyle p\displaystyle =\displaystyle \dfrac{13}{70} - \dfrac{2q}{7}Divide both sides by -7

We can now substitute the value of p to Equation 2 to solve for q.

\displaystyle -21p-10q\displaystyle =\displaystyle -\dfrac{9}{10}Write Equation 2
\displaystyle -21\left(\dfrac{13}{70}+\dfrac{2q}{7}\right)-10q\displaystyle =\displaystyle -\dfrac{9}{10}Substitute p
\displaystyle -\dfrac{39}{10}-6q+10q\displaystyle =\displaystyle -\dfrac{9}{10}Multiply both sides by -7
\displaystyle 4q\displaystyle =\displaystyle \dfrac{30}{10}Add like terms
\displaystyle 4q\displaystyle =\displaystyle 3Simplify the fraction
\displaystyle q\displaystyle =\displaystyle \dfrac{3}{4}Divide both sides by 4
b

Now solve for p.

Worked Solution
Create a strategy

Substitute the value of q from part (a) into Equation 1.

Apply the idea
\displaystyle -7p+2q\displaystyle =\displaystyle -\dfrac{13}{10}Write Equation 1
\displaystyle -7p-2\times \dfrac{3}{4}\displaystyle =\displaystyle -\dfrac{13}{10}Substitute q
\displaystyle -7p+\dfrac{3}{2}\displaystyle =\displaystyle -\dfrac{13}{10}Simplify fraction
\displaystyle -7p\displaystyle =\displaystyle -\dfrac{13}{10}-\dfrac{3}{2}Subtract \dfrac{3}{2} from both sides
\displaystyle -7p\displaystyle =\displaystyle -\dfrac{28}{10}Evaluate the fraction
\displaystyle p\displaystyle =\displaystyle \dfrac{2}{5}Divide both side by -7

Example 3

A man is five times as old as his son. Six years ago the man was ten times as old as his son.

We want to find their present ages

a

Use the fact that the man is four times as old as his son to set up Equation 1.

Worked Solution
Create a strategy

Let x represent the present age of the man, and y represent the present age of the son.

Apply the idea

The man is five times as old as his son, so x is equal to 5 times y.

\displaystyle x\displaystyle =\displaystyle 5yEquation 1
b

Use the fact that four years ago the man was nine times as old as his son to set up Equation 2.

Write the equation in the form ax+by=c, where a is positive.

Worked Solution
Create a strategy

To find their ages 4 years ago, we need to subtract 4 from x and y.

Apply the idea

We can express the age of the man from six years ago as x-4 and we can express the age of the son as y-4.

At this time the age of the man is ten times the age of the son, so we get the equation:

\displaystyle x-4\displaystyle =\displaystyle 9\left(y-4\right)Equation 2
\displaystyle x-4\displaystyle =\displaystyle 9y-36Expand the brackets
\displaystyle x\displaystyle =\displaystyle 9y-32Add 4 to both sides
\displaystyle x-9y\displaystyle =\displaystyle -32Subtract 9y from both sides
c

Solve for y to find the age of the son.

Worked Solution
Create a strategy

Substitute Equation 1 into Equation 2.

Apply the idea
\displaystyle x-9y\displaystyle =\displaystyle -32Write Equation 2
\displaystyle 5y-9y\displaystyle =\displaystyle -32Substitute x=5y
\displaystyle -4y\displaystyle =\displaystyle -32Simplify like terms
\displaystyle y\displaystyle =\displaystyle 8Divide both sides by -4

The son is 8 years old.

d

Solve for x to find the age of the man.

Worked Solution
Create a strategy

Substitute the value of y into Equation 1.

Apply the idea
\displaystyle x\displaystyle =\displaystyle 5yWrite Equation 1
\displaystyle x\displaystyle =\displaystyle 5 \times 8Substitute y=8
\displaystyle x\displaystyle =\displaystyle 40Evaluate

The man is 40 years old.

Idea summary

To solve two equations simultaneously using the substitution method, we may need to rearrange one of the equations so that one variable is the subject before we can perform the substitution.

Outcomes

ACMNA237

Solve linear simultaneous equations, using algebraic and graphical techniques, including using digital technology

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