Simultaneous equations are two equations with two unknowns (usually x and y) that are both true at the same time. Solving them means finding the one pair of values that fits both at once — like working out the price of a single apple and a single banana when you only know the totals of two mixed baskets.
Why this matters: real problems often lock two unknowns together — two products with two totals, a mixture of coins, choosing between two pricing plans. One equation alone isn't enough to pin both down; you need two, and simultaneous equations are the tool that cracks them. Graphically, the answer is the single point where the two lines cross.
The elimination method
The idea: combine the two equations so that one letter cancels out, leaving a simple equation in the other. Once you know one value, the rest follows.
- Make the number in front of one letter match in both equations.
- Add or subtract the equations to make that letter disappear.
- Solve for the letter that's left, then substitute back to find the other.
When the matching terms have the same sign, subtract the equations; when they have different signs, add. (Remember: "Same Signs Subtract / Different Add".)
x + y = 10 and x − y = 4: the y terms are +y and −y (different signs) → add the equations: 2x = 14, so x = 7. Substitute back: 7 + y = 10 → y = 3. Answer: x = 7, y = 3.
2x + y = 11 and x + y = 7: the y terms are both +y (same sign) → subtract: (2x − x) = (11 − 7), so x = 4. Substitute: 4 + y = 7 → y = 3.
| Matching terms | Signs | Do this |
|---|---|---|
| +y and −y | different | ADD (they cancel) |
| +y and +y | same | SUBTRACT (they cancel) |
| 2x and 2x | same | SUBTRACT |
Common mistake: when you subtract equations, subtract every term, including the numbers on the right — and take care with the signs. It's also easy to stop after finding x: always substitute back to find y too, and check both original equations hold.
In a nutshell: match one letter's coefficient, then add or subtract to eliminate it (same signs subtract, different signs add), solve for the survivor, and substitute back for the other.