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.

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.

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.

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 $+5x$).

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 $4x + 6y = 7$.

Right: Multiply every term: $2 \times (2x + 3y = 7)$ becomes $4x + 6y = 14$.

The Elimination Method

+5 XP

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.

Line the equations up, subtract to lose x, solve for y, then substitute back for x.

Text description

Top box: 5x + 3y = 23 above 5x − 2y = 8.
An arrow labelled same sign → subtract leads to 5y = 15, y = 3.
An arrow labelled substitute y = 3 leads to x = 2.8.

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.

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.

+5x and +5x: subtract. +4y and −4y: add.

Text description

The top box asks: same sign?
An arrow labelled yes leads left to Subtract; an arrow labelled no leads right to Add.
Both lead down to the strip: Goal: make the variable disappear.

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.

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: × 3 and × 2 make both x-coefficients 6, then subtract.

Text description

Top box: 2x + 3y = 14 above 3x + 5y = 22.
An arrow labelled LCM of 2 and 3 is 6 leads to 6x + 9y = 42 (the first equation × 3) and 6x + 10y = 44 (the second × 2).
An arrow labelled subtract leads to y = 2, then x = 4.

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.

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 method that fits the form of the equations.

Text description

Left column, Substitution: variable isolated; coefficient is 1; equations with fractions.
Right column, Elimination: standard form; coefficients match; same structure.
The strip reads: Both methods always work. Choose the cleaner path.

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.

Worked Example 1, Subtracting Equal Coefficients

Q.Solve $$\begin{cases} 5x + 3y = 23 \\ 5x - 2y = 8 \end{cases}$$

Solution

Identify: The $x$ coefficients are equal (both 5). Same sign → subtract.

Subtract Equation 2 from Equation 1: $(5x + 3y) - (5x - 2y) = 23 - 8$

$5y = 15$ → $y = 3$

Substitute back: $5x + 3(3) = 23$ → $5x = 14$ → $x = \dfrac{14}{5} = 2.8$

Check: Equation 2: $5(2.8) - 2(3) = 14 - 6 = 8$ ✓

Answer: $x = 2.8,\; y = 3$

Worked Example 2, Adding Opposite Coefficients

Q.Solve $$\begin{cases} 3x + 4y = 18 \\ 2x - 4y = 2 \end{cases}$$

Solution

Identify: The $y$ coefficients are opposites ($+4$ and $-4$). Opposite signs → add.

Add the equations: $(3x + 4y) + (2x - 4y) = 18 + 2$

$5x = 20$ → $x = 4$

Substitute back: $3(4) + 4y = 18$ → $12 + 4y = 18$ → $4y = 6$ → $y = 1.5$

Check: Equation 2: $2(4) - 4(1.5) = 8 - 6 = 2$ ✓

Answer: $x = 4,\; y = 1.5$

Worked Example 3, Multiplying to Match Coefficients

Q.Solve $$\begin{cases} 2x + 3y = 14 \\ 3x + 5y = 22 \end{cases}$$

Solution

Strategy: Eliminate $x$. LCM of 2 and 3 is 6. Multiply Eq 1 by 3, Eq 2 by 2.

Multiply: $\begin{cases} 6x + 9y = 42 \\ 6x + 10y = 44 \end{cases}$

Subtract: $(6x + 10y) - (6x + 9y) = 44 - 42$ → $y = 2$

Back-substitute: $2x + 3(2) = 14$ → $2x = 8$ → $x = 4$

Check: Eq 2: $3(4) + 5(2) = 12 + 10 = 22$ ✓

Answer: $x = 4,\; y = 2$

Brain Trainer

4 quick-fire drills. Beat the clock.

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 $3x + 5y = 22$?

Practice

5 MCQs and 3 short-answer questions. Target: 80% accuracy.

To eliminate $x$ from $\begin{cases} 3x + 2y = 10 \\ 3x - 5y = 4 \end{cases}$, you should:

To eliminate $y$ from $\begin{cases} 2x + 3y = 11 \\ 5x - 3y = 10 \end{cases}$, you should:

What is the first step in solving $\begin{cases} 2x + 3y = 13 \\ 4x + 5y = 23 \end{cases}$ by elimination?

The solution to $\begin{cases} x + y = 7 \\ x - y = 3 \end{cases}$ is:

Which method is most efficient for $\begin{cases} y = 2x + 1 \\ 3x + 4y = 23 \end{cases}$?

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.

$(5x + 2y) - (3x + 2y) = 24 - 16$

$2x = 8$ → $x = 4$

Substitute back: $3(4) + 2y = 16$ → $12 + 2y = 16$ → $2y = 4$ → $y = 2$

Check Eq 1: $5(4) + 2(2) = 20 + 4 = 24$ ✓. Check Eq 2: $3(4) + 2(2) = 12 + 4 = 16$ ✓.

Answer: $x = 4,\; y = 2$

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.

Your answer:

Correct multipliers chosen

Correct elimination performed

$x = 1$

$y = 5$

Model Answer

Eliminate $x$. LCM of 2 and 3 is 6.

Equation 1 $\times$ 3: $6x + 9y = 51$

Equation 2 $\times$ 2: $6x + 10y = 56$

Subtract: $(6x + 10y) - (6x + 9y) = 56 - 51$ → $y = 5$

Back-substitute: $2x + 3(5) = 17$ → $2x = 2$ → $x = 1$

Check: $3(1) + 5(5) = 3 + 25 = 28$ ✓.

Answer: $x = 1,\; y = 5$

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: $\begin{cases} l + w = 120 \\ l - w = 20 \end{cases}$

Add: $2l = 140$ → $l = 70$ m

$w = 120 - 70 =$ $50$ m

(c) Area $= 70 \times 50 =$ $3500$ m²

Verify perimeter: $2(70) + 2(50) = 140 + 100 = 240$ m ✓

Answer: Length 70 m, width 50 m, area 3500 m²

Review

Consolidate and reflect before moving on.

Stretch Challenge

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?

I have completed this lesson and checked my answers.

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Simultaneous Equations Elimination

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