Year 11 General Mathematics
Simultaneous Equations: Substitution and Elimination
Two equations. Two unknowns. One ordered pair that makes both of them work.
Getting started
One unknown is usually enough trouble. Today, we are dealing with two.
In this lesson, you will solve problems where two unknown values are connected by two equations. Your job is to find the pair of values that makes both equations true.
For example, if the unknowns are x and y, the final answer will usually look like this:
This pair is called the solution.
One equation
Many pairs work, such as (6, 4) and (8, 2).
Two equations together
Now the pair must work in both equations. The pair (6, 4) works.
Checkpoint
Check that (6, 4) works in both equations above. Write one line for each equation.
Substitution
Substitution means replacing one thing with another thing that has the same value. It is usually the easiest method when one equation already has x or y by itself.
What substitution is doing
If an equation says y = x + 4, then wherever you see y, you are allowed to write x + 4 instead. That turns two equations with two unknowns into one equation with one unknown.
- Find the equation where one variable is already by itself.
- Copy the other equation.
- Replace the isolated variable with the expression it equals.
- Solve the new one-variable equation.
- Put that answer back into either original equation to find the second value.
- Check both original equations.
Worked example Substitution when y is already by itself
Solve:
Start with the equation that has y by itself:
This means y and x + 4 are equal. In the second equation, replace y with x + 4.
Now there is only x, so solve it like a normal linear equation.
Use x = 2 to find y.
Answer: (2, 6).
Check the answer
Your turn
Solve using substitution. Show the replacement step clearly.
Check your solution
Elimination
Elimination means making one variable disappear by adding or subtracting whole equations. It is usually the easiest method when the equations are already lined up, especially when one variable has matching or opposite coefficients.
What elimination is doing
If one equation has +y and another has -y, adding the equations makes the y terms cancel. If both equations have the same +2y, subtracting the equations makes the 2y terms cancel.
- Line up the x terms, y terms and numbers.
- Look for a variable that can cancel.
- Add if the signs are opposite, such as +y and -y.
- Subtract if the signs are the same, such as +2y and +2y.
- If nothing cancels yet, multiply one or both whole equations first.
- Solve, substitute back, and check.
Worked example Opposite terms are ready to cancel
Solve:
The equations contain +y and -y. Add the equations and the y terms cancel.
Substitute x = 6 into one original equation.
Answer: (6, 3).
Worked example Multiply one equation first
Solve:
The first equation has y. The second equation has 2y. Multiply the whole first equation by 2 so both equations have 2y.
Now subtract the second equation because both equations have +2y.
Substitute x = 2 into the first original equation.
Answer: (2, 3).
Your turn
Solve using elimination.
Check your solution
Choosing a method
Both methods can give the same answer. The trick is choosing the method that makes the work easier for the equations in front of you.
A decision making table
| What you see | Try this | Why |
|---|---|---|
| Already says x = ... or y = ... | Substitution | One variable is ready to replace. |
| Easy to make x = ... or y = ... | Substitution | Rearrange first, then replace. |
| Opposite terms, like +y and -y | Elimination: add | Opposites make zero. |
| Matching terms, like +2y and +2y | Elimination: subtract | Same terms subtract to zero. |
| Nothing cancels yet | Elimination: multiply first | Make a matching term, then cancel. |
Example: choose substitution
Use substitution when one equation already has a variable by itself.
Since y already equals x + 4, replace y in the second equation.
Example: choose elimination
Use elimination when the equations are lined up and a variable can cancel.
The +y and -y cancel if you add the equations.
Example: multiply first
Sometimes nothing cancels straight away.
The first equation has y. The second has 2y. Multiply the first equation by 2 to make the y terms match.
Practice
Open one section at a time. For each question, choose a method, solve the equations, and check your answer in at least one original equation.
Practice tools
Substitution practice
Use substitution. Each question has one variable already by itself.
Check answers
Elimination practice
Use elimination. One variable can be cancelled by adding or subtracting the equations.
Check answers
Elimination with multiplying first
Use elimination, but multiply one equation first so one variable can cancel.
Check answers
Mixed method practice
Choose substitution or elimination. Write why you chose that method before solving.
Check answers
Extension challenge
First, solve a tougher pair of equations. Then make your own pair and check that your chosen solution works in both.
Part A: solve
Solve this pair of equations. You may use substitution or elimination, but you should explain why your method makes sense.
Part B: create
Create two new pairs of simultaneous equations with the same solution as Part A.
- Make one pair that is best solved by substitution.
- Make one pair that is best solved by elimination.
- Check that your solution works in both pairs.
Extension answer
Exit ticket
Complete these before the end of the lesson.
Explain
In your own words, what does the solution to simultaneous equations represent?
Solve by substitution
Solve by elimination
Exit ticket answers
- The solution is the ordered pair that makes both equations true at the same time.
- Substitution problem: (5, 7).
- Elimination problem: (4, 3).