When neither variable is isolated, elimination lets you add or subtract entire equations to make a variable disappear, leaving one equation with one unknown.
45 min 5 MCQ + 3 SAQ MAS-EQN-C-02
Think First
Before we begin: If $2x + 3y = 13$ and $2x - y = 5$, notice that both equations have $2x$. What would happen if you subtracted one equation from the other? Try to work out what $y$ would be.
Come back to this after you have worked through the lesson.
1
The Big Idea
Elimination is powerful when both equations are in standard form ($ax + by = c$) and neither variable has a coefficient of 1. By making the coefficients of one variable match, you can add or subtract the equations to eliminate that variable entirely. This leaves a single equation with one unknown that you can solve directly.
Know
The elimination method for solving simultaneous equations. When to add and when to subtract equations.
Understand
Why matching coefficients allows a variable to be eliminated. How to multiply equations to create matching coefficients.
Can Do
Solve simultaneous equations by elimination. Decide whether substitution or elimination is more efficient.
2
Lesson Objectives
Identify when coefficients already match for elimination.
Multiply one or both equations to create matching coefficients.
Correctly add or subtract equations to eliminate a variable.
Choose the most efficient method (substitution or elimination) for a given pair of equations.
3
Key Vocabulary
Elimination
A method for solving simultaneous equations by adding or subtracting equations to remove one variable.
Matching coefficients
Making the coefficients of a variable equal (or opposite) in both equations.
Add equations
Used when the coefficients of the variable to eliminate are opposites (e.g. $+3y$ and $-3y$).
Subtract equations
Used when the coefficients of the variable to eliminate are equal (e.g. $+5x$ and $+5x$).
4
Spot the Trap
Wrong: Adding equations when coefficients are equal. In $3x + 2y = 12$ and $3x + 5y = 21$, adding gives $6x + 7y = 33$, which does not eliminate anything.
Right: When coefficients are equal, subtract the equations: $(3x + 5y) - (3x + 2y) = 21 - 12$ gives $3y = 9$.
Wrong: Forgetting to multiply every term when scaling an equation. Multiplying $2x + 3y = 7$ by 2 to get $4x + 6y = 7$.
The elimination method has six steps: line up both equations with $x$ terms, $y$ terms, and constants aligned. Match coefficients for one variable by multiplying one or both equations. Add or subtract the equations to eliminate that variable. Solve the resulting equation. Substitute back to find the other variable. Check both values in both original equations.
Same sign → subtract. Opposite signs → add. Then solve and check.
make a variable disappear
Line up first
Write both equations with x terms, y terms, and constants aligned.
Match one variable
Choose the variable that needs the smallest multiplier to match coefficients.
Verify both equations
A solution only counts if it satisfies BOTH original equations.
6
Add or Subtract?
+5 XP
The decision to add or subtract depends on the signs of the matching coefficients. If the coefficients have the same sign (both positive or both negative), subtract the equations. If they have opposite signs, add the equations. The goal is always to make the target variable disappear.
+5x and +5x → subtract. +4y and -4y → add. Gone.
same sign = subtract
Same sign → subtract
Both +3y: subtract to get 0y. Both -2x: subtract to get 0x.
Opposite signs → add
+4y and -4y: add to get 0y. +5x and -5x: add to get 0x.
Check your choice
After combining, one variable should be completely gone. If not, you added/subtracted wrong.
7
Multiplying to Match
+5 XP
When no coefficients already match, multiply one or both equations by numbers that create matching coefficients. Use the LCM (lowest common multiple) of the coefficients to decide what to multiply by. Remember to multiply every term on both sides of the equation.
2x and 3x → LCM is 6. Eq 1 x3, Eq 2 x2. Then subtract.
multiply every term
Use the LCM
For coefficients 2 and 3, the LCM is 6. Multiply one equation by 3 and the other by 2.
Multiply EVERY term
When scaling, multiply the constant too. 2(2x + 3y = 7) gives 4x + 6y = 14.
Pick the easier variable
Choose whichever variable needs smaller multipliers to match.
8
Substitution vs Elimination
+5 XP
Both methods solve the same problems, but one is usually more efficient depending on the form of the equations. Use substitution when a variable is already isolated or has coefficient 1. Use elimination when both equations are in standard form with no easy isolation.
Isolated variable → substitution. Standard form → elimination. Both work.
pick the faster path
Substitution for isolation
If you see $y = 3x + 2$, substitution is the natural choice.
Elimination for standard form
If both equations are $ax + by = c$, elimination is usually faster.
Both give the same answer
The choice is about efficiency, not correctness. Either method works.
If $3x + 2y = 10$ and $3x - 5y = 4$, what operation eliminates $x$?
Subtract
If $2x + 3y = 11$ and $5x - 3y = 10$, what operation eliminates $y$?
Add
Solve: $x + y = 7$ and $x - y = 3$
x = 5, y = 2
What do you multiply $2x + 3y = 14$ by to match $x$ with $3x + 5y = 22$?
3
Practice
5 MCQs and 3 short-answer questions. Target: 80% accuracy.
1.To eliminate $x$ from $$\begin{cases} 3x + 2y = 10 \\ 3x - 5y = 4 \end{cases}$$, you should:
Correct! The $x$ coefficients are equal (both 3) with the same sign. Subtracting eliminates $x$: $(3x + 2y) - (3x - 5y) = 10 - 4$ gives $7y = 6$.
Not quite. Look at the $x$ coefficients: both are 3. When coefficients are equal and have the same sign, you subtract to eliminate.
2.To eliminate $y$ from $$\begin{cases} 2x + 3y = 11 \\ 5x - 3y = 10 \end{cases}$$, you should:
Correct! The $y$ coefficients are opposites ($+3$ and $-3$). Adding eliminates $y$: $(2x + 3y) + (5x - 3y) = 11 + 10$ gives $7x = 21$.
Look at the $y$ coefficients: $+3$ and $-3$. When coefficients have opposite signs, you add to eliminate.
3.What is the first step in solving $$\begin{cases} 2x + 3y = 13 \\ 4x + 5y = 23 \end{cases}$$ by elimination?
Correct! The $x$ coefficients are 2 and 4. Multiplying Equation 1 by 2 gives $4x + 6y = 26$, which matches the $4x$ in Equation 2. Then you can subtract.
Before adding or subtracting, the coefficients must match. The $x$ coefficients are 2 and 4. What do you multiply Equation 1 by to make the $x$ terms equal?
4.The solution to $$\begin{cases} x + y = 7 \\ x - y = 3 \end{cases}$$ is:
Correct! Add the equations: $2x = 10$ so $x = 5$. Substitute back: $5 + y = 7$ so $y = 2$.
The $y$ coefficients are opposites ($+1$ and $-1$), so add the equations. Then back-substitute to find the other variable.
5.Which method is most efficient for $$\begin{cases} y = 2x + 1 \\ 3x + 4y = 23 \end{cases}$$?
Correct! The first equation already has $y$ isolated ($y = 2x + 1$). Substitution is the natural choice here — substitute directly into the second equation.
Look at the form of the equations. When one variable is already isolated, substitution is usually faster than elimination.
6.Solve by elimination: $$\begin{cases} 5x + 2y = 24 \\ 3x + 2y = 16 \end{cases}$$. Show all steps and verify your solution.
Your answer:
Correct elimination operation chosen
$x = 4$
$y = 2$
Both equations verified
Model Answer
The $y$ coefficients are equal (both 2). Same sign → subtract Equation 2 from Equation 1.
7.Solve by elimination: $$\begin{cases} 2x + 3y = 17 \\ 3x + 5y = 28 \end{cases}$$. You will need to multiply both equations. Show all working clearly.
8.A rectangular paddock has perimeter 240 m. The length is 20 m more than the width. (a) Define variables and set up two equations. (b) Solve using elimination to find the dimensions. (c) Calculate the area and verify your dimensions satisfy the original perimeter.
Your answer:
Variables defined and two equations set up (1 mark)
Correct elimination method used (2 marks)
Area calculated correctly (1 mark)
Perimeter verified (1 mark)
Model Answer
(a) Let $l$ = length (m), $w$ = width (m).
Perimeter: $2l + 2w = 240$
Length-width: $l = w + 20$ or $l - w = 20$
(b) Simplify perimeter: divide by 2 → $l + w = 120$
Now solve: $\begin{cases} l + w = 120 \\ l - w = 20 \end{cases}$
Solve by elimination: $$\begin{cases} 3x + 4y = 26 \\ 5x - 2y = 18 \end{cases}$$. You will need to multiply one equation before eliminating. Show every step clearly. (Hint: which variable needs the smaller multiplier?)
Key Idea
Elimination: line up equations, match coefficients, add or subtract to eliminate a variable, solve, back-substitute, and verify both equations.
Common Trap
Adding equations when coefficients are equal (instead of subtracting). Same sign always means subtract. Opposite signs always means add.
Connection
Substitution and elimination solve the same problems. Substitution wins when a variable is isolated; elimination wins when both equations are in standard form.
Interactive: Elimination Step Builder — drag and drop the steps of elimination into the correct order.
Elimination Expert
Method Chooser
Word Problem Solver
Daily Challenge
Solve: $4x + 3y = 23$ and $4x - y = 11$ by elimination. Time yourself — can you do it in under 60 seconds?
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