Year 9: Simultaneous equations

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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:

 $1) 2 x + y = 72) x - y = 3$

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.

 $1) x = y + 2 (rearranged from x - y = 2) 2) 3 x + 2 y = 10 Substitute (x = y + 2) into 2) : 3 (y + 2) + 2 y = 103 y + 6 + 2 y = 105 y + 6 = 105 y = 4 y = 4 / 5 = 0.8 x = 0.8 + 2 = 2.8 Solution : (x, y) = (2.8, 0.8)$

4. Elimination

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

 $1) 2 x + 3 y = 122) 2 x - y = 6 Subtract equation (2) from (1) : (2 x + 3 y) - (2 x - y) = 12 - 62 x + 3 y - 2 x + y = 64 y = 6 y = 6 / 4 = 1.5 Substitute y = 1.5 into (2) : 2 x - 1.5 = 62 x = 7.5 x = 3.75 Solution : (x, y) = (3.75, 1.5)$

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.