Learning Objectives
You should be able to:
- Explain what simultaneous equations are.
- Identify the unknowns shared by two equations.
- Solve simultaneous equations using elimination.
- Solve simultaneous equations using substitution.
- Choose a suitable method for a question.
- Check that your answers satisfy both equations.
Introduction
This lesson develops Simultaneous Equations from meaning and patterns before moving to calculations. Look for what the quantities, shapes or symbols represent, then connect that idea to the method.
Key Notes
Read the definitions and key facts below carefully. Each rule is most useful when you can explain why it fits the situation, not simply recall it.
🎬 Two equations, one meeting point
The solution satisfies both equations at the same time.
🤔 Think About This...
Imagine a tuckshop sells apples and bananas.
One customer buys:
- 2 apples
- 1 banana
The total cost is $8.
Another customer buys:
- 1 apple
- 2 bananas
The total cost is $7.
Let:
The two purchases give us:
📖 What are Simultaneous Equations?
Simultaneous equations are two or more equations that contain the same unknown variables.
They are solved together because the values of the unknowns must make every equation true at the same time.
Both equations contain x and y.
🧠 The StudyNest Thinking Method
👉 Which method will make this question easier?
There are two common algebraic methods:
- Elimination
- Substitution
Both methods can lead to the same correct answer. The best method depends on how the equations are written.
🎯 Choosing the Best Method
In many questions, either method will work. However, one method may be quicker or easier than the other.
| If you notice... | A useful method to try | Why? |
|---|---|---|
| One variable already stands alone, such as y = x + 2 | Substitution | You can replace that variable immediately. |
| One variable can be made the subject in one simple step | Substitution | Rearranging and replacing the variable will be straightforward. |
| A pair of terms already has opposite signs, such as +y and −y | Elimination | The variable disappears when the equations are added. |
| A pair of variables has the same coefficient, such as +y and +y | Elimination | The variable disappears when one equation is subtracted from the other. |
| The coefficients can easily be made equal, such as 2x and 4x | Elimination | Multiplying one equation creates matching coefficients. |
Spend a few seconds looking at both equations before you begin. A little planning can save several steps.
Unless the question tells you which method to use, you may normally use any correct method.
If the question specifically says “Use elimination” or “Use substitution”, use the requested method and show your working clearly.
🤔 Which Method Would You Choose?
Do not solve these yet. Simply decide which method looks easier.
y = 2x + 5
x + y = 11
Suggested method: Substitution
x + y = 9
x − y = 5
Suggested method: Elimination
2x + y = 8
x + y = 5
Suggested method: Elimination
3x = y + 7
2x + y = 11
Suggested method: Substitution after making one variable the subject
➕ Method 1: Elimination
The word eliminate means to remove.
In this method, we add or subtract the equations so that one variable disappears. This leaves an equation containing only one unknown.
1. Line up the equations.
2. Look for equal or opposite variable terms.
3. Add or subtract the equations.
4. Solve for the remaining variable.
5. Substitute back to find the second variable.
6. Check both answers.
Example 1: Opposite Terms
Solve:
🔍 What do you notice?
One equation contains +y and the other contains −y.
These are opposite terms.
Add the equations:
The y-terms cancel:
Substitute x = 5 into one of the original equations:
Example 2: Equal Terms
Solve:
🔍 What do you notice?
Both equations contain +y.
Because the y-terms are equal, subtract the second equation from the first:
Substitute x = 3 into:
✖️ When the Coefficients Do Not Match
Sometimes no variable can disappear immediately.
In that situation, multiply one or both equations so that one pair of coefficients becomes equal.
Example 3
Solve:
The first equation contains x, while the second contains 2x.
Multiply the whole first equation by 2:
The equations are now:
The x-terms are now equal, so subtract the first equation from the second:
Substitute y = 4 into:
🔄 Method 2: Substitution
The word substitute means to replace one thing with another equal thing.
In simultaneous equations, substitution means replacing one variable with an equivalent expression.
1. Make one variable the subject if necessary.
2. Substitute that expression into the other equation.
3. Solve the new equation.
4. Substitute back to find the second variable.
5. Check both answers.
Example 4: One Variable is Already the Subject
Solve:
🔍 What does y equal?
The first equation tells us:
Replace y with x + 1 in the second equation:
Substitute x = 4 into:
Example 5: No Variable is Initially the Subject
Solve using substitution:
Neither variable is currently alone.
Choose the simpler equation:
Make x the subject by subtracting y from both sides:
Substitute 6 − y for x in the first equation:
Expand and simplify:
Substitute y = 3 into:
🌍 Real-Life Example
A tuckshop sells juice and chips.
Two juices and one packet of chips cost $9.
One juice and two packets of chips cost $12.
Let:
The equations are:
Either method can be used, but elimination is convenient here because the coefficients can easily be made equal.
🕵️ Checking Your Answers
A correct pair of values must satisfy both original equations.
Suppose:
Check the first equation:
Check the second equation:
📈 Looking Ahead
Each equation is drawn as a graph, and the point where the two graphs intersect gives the solution.
We will study this method in the Graphs topic.
⚠ Common Mistakes
Adding the equations when equal terms should be subtracted.
Look carefully at the signs and coefficients before choosing an operation.
Multiplying only one term when trying to create matching coefficients.
Multiply every term and the value on the other side of the equation.
Forgetting to substitute back after finding the first variable.
Substituting into the same equation incorrectly. Use brackets when replacing an expression.
Checking only one equation. Both original equations must be true.
Practice Questions
First decide which method looks easier. Then solve.
-
x + y = 10
x − y = 4 -
2x + y = 14
x + y = 9 -
y = x + 2
x + y = 11 -
x + y = 12
2x − y = 6 -
2x + y = 11
x + 2y = 10 -
3x + y = 13
x + y = 7
✅ Show Answers
- x = 7, y = 3
- x = 5, y = 4
- x = 4.5, y = 6.5
- x = 6, y = 6
- x = 4, y = 3
- x = 3, y = 4
🎯 Exam Tip
If a method is named in the question, use that method.
Always show enough working to make your method clear.
🧠 Did You Know?
⭐ What You Can Do Now
- ✅ I know what simultaneous equations are.
- ✅ I understand what elimination means.
- ✅ I can solve equations using elimination.
- ✅ I understand what substitution means.
- ✅ I can make a variable the subject before substituting.
- ✅ I can choose a suitable method.
- ✅ I can check my answers in both equations.
🎉 Well Done!
You can now solve simultaneous equations strategically instead of choosing a method at random.
Look at the equations → Choose a method → Solve → Substitute back → Check
🧠 Let’s Think Together
Before calculating, identify the information given, the result required and the mathematical relationship that connects them. Predict whether the answer should be larger, smaller or unchanged, then use that prediction to check the result.
📈 Your Progress
Lesson 12 of 22
Exam Focus
Show the mathematical relationship you are using, keep notation consistent and give a clear final answer. Use a quick estimate, diagram or inverse operation to check that the result is reasonable.
Summary
- Understand the meaning of each idea before selecting a method.
- Use definitions, properties and notation consistently.
- Show working and check that the final result is sensible.