Learn: Simultaneous Equations

Review the key ideas, then practise whenever you’re ready.

The idea

One clue with two unknowns - like “two numbers add to 6” ( $x + y = 6$ ) - has lots of answers. Add a second clue at the same time - “their difference is 2” ( $x - y = 2$ ) - and only ONE pair in the world fits both clues ( $x = 4, y = 2$ ). That is what simultaneous equations do: two clues lock in two unknowns.

Your three solving moves:

1
Add to Eliminate (Opposite Signs) - when a letter has opposite signs (like $+y$ and $-y$ ), add the equations so that letter cancels to $0$ .
2
Subtract to Eliminate (Same Signs) - when a letter has the exact same term (like $+2y$ and $+2y$ ), subtract one equation from the other.
3
Substitute (Swap a Bracket) - when one equation already starts with $y = \dots$ , swap that expression in place of $y$ in the other equation.

Quick memory tips

Memory rule: Opposite signs → ADD; Same signs → SUBTRACT
If no numbers match yet, multiply one whole equation (both sides!) so a letter matches
Always check your $(x, y)$ answer in BOTH original equations

One Point That Makes Both True

One clue ( $x + y = 6$ ) has lots of answers: $(1,5), (2,4), (3,3), (4,2)\dots$ A second clue ( $x - y = 2$ ) locks in the single crossing point (4, 2) where both lines agree!

4 + 2 = 6

✓

4 - 2 = 2

✓

Interactive Strategy Picker: Which Move Should You Use?

\begin{cases} 2x + y = 9 \\ 3x - y = 6 \end{cases} \quad\xrightarrow{\text{ADD}}\quad 5x = 15 \;\Rightarrow\; x = 3

Because +y and −y have opposite signs, adding the equations cancels $y$ to 0 immediately.

Worked example

Every step explained - why we do it, and what happens.

1
$\begin{cases} 2x + y = 9 \\ 3x - y = 6 \end{cases}$
Why: Scan both equations before doing any arithmetic: look for a letter with matching numbers.
What happens: The top equation has +y and the bottom has −y - opposite signs!
2
$(2x + y) + (3x - y) = 9 + 6$
Why: Opposite signs (+y and −y) mean we ADD the two equations straight down.
What happens: +y and −y cancel to 0, so y disappears completely.
3
$5x = 15 \;\Rightarrow\; x = 3$
2x + 3x = 5x and 9 + 6 = 15. Divide by 5 to get x = 3.
4
$2(3) + y = 9 \;\Rightarrow\; y = 3$
Why: Plug x = 3 back into the first equation (2x + y = 9) to unlock y.
What happens: 6 + y = 9, so y = 3.
5
$3(3) - 3 = 9 - 3 = 6$
Why: The check: test x = 3 and y = 3 in the SECOND equation (3x − y = 6).
What happens: 9 − 3 = 6 ✓ - the two lines cross at (3, 3).

Scaling Equations & Curve Intersections (Optional Toolbox)

What to do when no coefficients match yet, or when a straight line crosses a parabola.

Levels 3–6

1. Scaling to Make Twins

If you have $x + 2y = 10$ and $3x - 4y = 5$ , multiply the entire first equation by 2 ( $2x + 4y = 20$ ). Now $+4y$ and $-4y$ match!

2. Line Meets Parabola

When $y = x^2 - 2x$ meets $y = x + 4$ , set the right sides equal ( $x^2 - 2x = x + 4$ ), move everything to one side ( $= 0$ ), and factor!

Real-world example

At the snack stall, 1 taco and 1 smoothie cost 9 coins. The next customer buys 2 tacos and 3 smoothies for 21 coins. How much is 1 smoothie, and how much is 1 taco?

1
$\begin{cases} T + S = 9 \\ 2T + 3S = 21 \end{cases}$
Why: Let T be the taco price and S be the smoothie price. Each sentence gives one equation.
What happens: Two clues, two unknowns.
2
$2T + 2S = 18$
Why: Double the entire first order (multiply every term of T + S = 9 by 2) so both equations have 2T.
What happens: 2 tacos + 2 smoothies cost 18 coins.
3
$(2T + 3S) - (2T + 2S) = 21 - 18 \;\Rightarrow\; S = 3$
Why: Subtract the two orders: 2T − 2T cancels out, leaving 1 extra smoothie = 3 coins.
What happens: 1 smoothie costs 3 coins.
4
$\text{Trap: } 2(T + S) = 9 \;\Rightarrow\; 2T + 2S = 9\,?$
Why: The glitch: doubling the left side (2T + 2S) but forgetting to double the 9 on the right side.
What happens: If you double the food, you must double the price: 2 × 9 = 18.
5
$T = 9 - 3 = 6, \qquad 2(6) + 3(3) = 21$
Why: The check: 1 taco is 6 coins, 1 smoothie is 3 coins, and 2(6) + 3(3) = 12 + 9 = 21 coins.
What happens: Smoothie = 3 coins, Taco = 6 coins ✓.

Common glitch

Three classic simultaneous-equation glitches:

1.Forgetting to multiply the right-hand side when scaling

Wrong:Doubling $T + S = 9$ and writing $2T + 2S = 9$

Right:Multiply every term on BOTH sides of the equals sign: $2T + 2S = 18$ .

2.Adding when the signs are the SAME

Wrong:Adding $2x + 3y = 12$ and $x + 3y = 9$ gives $3x + 6y = 21$ ( $y$ did not cancel)

Right:Same signs ( $+3y$ and $+3y$ ) → SUBTRACT one equation from the other so $3y - 3y = 0$ .

3.Sign error when subtracting a negative term

Wrong:In $(3x + 2y) - (3x - y)$ , writing $2y - y = y$

Right:Subtracting a negative is adding: $2y - (-y) = 2y + y = 3y$ .

Linear Equation Solver - Solve the single-variable equation after eliminating or substituting.
Linear & Quadratic Graphing Calculator - Plot lines and parabolas to inspect their coordinates.

Next up: Sequences & Series →

Explore the next topic.

Ready to practise? Start with a warm-up, or return to the ladder.

Start Warm-Up Level →Back to Ladder

Learning is optional - the ladder is always open.

Loading math...

Common glitch

Three classic simultaneous-equation glitches:

1.Forgetting to multiply the right-hand side when scaling

Wrong:Doubling $T + S = 9$ and writing $2T + 2S = 9$

Right:Multiply every term on BOTH sides of the equals sign: $2T + 2S = 18$ .

2.Adding when the signs are the SAME

Wrong:Adding $2x + 3y = 12$ and $x + 3y = 9$ gives $3x + 6y = 21$ ( $y$ did not cancel)

Right:Same signs ( $+3y$ and $+3y$ ) → SUBTRACT one equation from the other so $3y - 3y = 0$ .

3.Sign error when subtracting a negative term

Wrong:In $(3x + 2y) - (3x - y)$ , writing $2y - y = y$

Right:Subtracting a negative is adding: $2y - (-y) = 2y + y = 3y$ .

Learn: Simultaneous Equations

The idea

Your three solving moves:

Quick memory tips

Worked example

Real-world example

Common glitch

Try a related calculator

Learn: Simultaneous Equations

The idea

Your three solving moves:

Quick memory tips

Worked example

Real-world example

Common glitch

Try a related calculator