Year 10: Simultaneous equations

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When two (or more) equations share the same variables, we call them simultaneous equations. Solving them means finding the values that satisfy all equations at once. In Year 10 you’ll focus on pairs of linear equations in two variables—perfect preparation for coordinate geometry, economics models, and senior algebra.

Work through the sections in order, then try the linked Year 10 skill quizzes after each section.

1. What does a solution look like?

For the system

$2x+3y=7x–y=1$

the solution is an ordered pair$(x,y)$. Here it’s $(2,1)$, because those numbers make both equations true.

2. Three possible outcomes

• One solution – lines cross at a single point (most common).
• No solution – lines are parallel (same gradient, different intercepts).
• Infinitely many solutions – lines coincide (identical equations in disguise).

3. Solving by graphing

1. Rewrite each equation in slope‑intercept form ($y=mx+c$).
2. Draw both lines on the same set of axes.
3. The intersection point is the solution.

Great for visual understanding, but accuracy depends on your drawing scale.

4. Solving by substitution

Best when one equation is already (or easily rearranged) into a single‑variable form.

$y=3x–5\left(1\right)2x+y=1\left(2\right)\text{Substitute}\left(1\right)\text{into}\left(2\right):2x+(3x–5)=15x–5=15x=6x=6/5=1.2\text{Back}\text{into}\left(1\right):y=3\left(1.2\right)–5=3.6–5=–1.4\text{Solution}:(1.2,–1.4)$

5. Solving by elimination

Ideal when coefficients line up neatly.

$3x+2y=14\left(1\right)2x–2y=2\left(2\right)\text{Add}\left(1\right)\text{and}\left(2\right):5x=16x=16/5=3.2\text{Substitute}\text{into}\left(2\right):2\left(3.2\right)–2y=26.4–2y=2–2y=–4.4y=2.2\text{Solution}:(3.2,2.2)$

6. Choosing a method

• Graphing – quick sketch or tech tool, good for sense‑checking.
• Substitution – one variable isolated.
• Elimination – coefficients easily matched by multiplying.

7. Check your answer

Plug your $(x,y)$ back into both original equations. If either fails, re‑trace your algebra—sign slips lurk!

Exercise

8. Common mistakes

• Arithmetic slips: Keep fractions exact until the end to avoid rounding errors.
• Forgetting to multiply both sides: In elimination, when you scale an equation, every term (including the constant) must be multiplied.
• Sign errors in substitution: Enclose substituted expressions in brackets.

9. Where simultaneous equations pop up

• Business: Break‑even analysis where revenue and cost lines meet.
• Science: Finding intersection of temperature vs time curves.
• Navigation: Determining position using intersecting bearings.

10. Summary

Simultaneous equations are a mathematical meeting point—find where two rules intersect and you unlock countless real‑world insights. Master graphing for intuition, substitution for precision, and elimination for speed, and you’ll have a solution strategy for any pair of lines that come your way.