Year 9: Simultaneous equations

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Work through the sections in order, then try the linked Year 9 skill quizzes after each section.

When you have two separate equations that share the same variables, you’re looking at what we call simultaneous equations. Solving them means finding a pair of values (for example, $x$ and $y$) that satisfy both equations at the same time. Think of it as figuring out the coordinates of a point where two straight lines meet.

1. What is a “solution”?

In a simple example with $x$ and $y$, you might have these two equations:

$1)2x+y=72)x–y=3$

A “solution” is a pair of numbers $(x,y)$ that makes both equations true at the same time.

2. Graph perspective

Each equation represents a straight line on an x–y graph:

• Where the lines intersect is the simultaneous solution $(x,y)$.
• If the lines never meet (parallel), there’s no solution.
• If the lines are actually the same line, then every point on that line is a solution (infinitely many).

3. Substitution

Handy when you can easily isolate one variable in one of the equations.

$1)x=y+2(\text{rearranged}\text{from}x–y=2\left)2\right)3x+2y=10\text{Substitute}(x=y+2)\text{into}2):3(y+2)+2y=103y+6+2y=105y+6=105y=4y=4/5=0.8x=0.8+2=2.8\text{Solution}:(x,y)=(2.8,0.8)$

4. Elimination

Great when you can line up variables to cancel one out by adding or subtracting equations.

$1)2x+3y=122)2x–y=6\text{Subtract}\text{equation}\left(2\right)\text{from}\left(1\right):(2x+3y)–(2x–y)=12–62x+3y–2x+y=64y=6y=6/4=1.5\text{Substitute}y=1.5\text{into}\left(2\right):2x–1.5=62x=7.5x=3.75\text{Solution}:(x,y)=(3.75,1.5)$

5. Checking your solution

After you find $(x,y)$, always plug it back into both equations:

• If it satisfies both, you’re correct.
• If not, look out for sign errors or arithmetic slips.

6. Possible outcomes

• One unique solution: The lines intersect at exactly one point.
• No solution: The lines are parallel and never meet.
• Infinitely many solutions: Both equations describe the same line.

7. Common pitfalls (and how to avoid them)

• Sign mix-ups: Subtracting one equation from another can lead to confusion with negatives.
• Forgetting to multiply all terms: In elimination, if you multiply an equation, apply it to every term (including constants!).
• Stopping too soon: Always check the solution in both original equations.