Mathematics 0580 · Syllabus tracker
Simultaneous
Solving two equations at once by elimination or substitution, forming them from problems, and one linear with one quadratic.
How to use this
Read each "I can…" statement and rate yourself honestly. Then try its quick check. If you get a check wrong on something you rated Got it, we'll suggest moving it to Shaky. My progress compares your ratings with your past-paper results to find topics you think you know but keep losing marks on.
Loading the tracker…
Quick revision
Simultaneous on one sheet
For the night before. The full notes teach the topic; this is just what you must remember.
At a glance
Solving two equations together: elimination, substitution, word problems, and line-and-curve pairs.
1Elimination
- Make the coefficient of one letter the same in both equations (multiply one or both equations).
- Same signs → subtract; different signs → add.
- Solve for the remaining letter, then substitute back to find the other.
- Check both values in the equation you didn't substitute into.
How to do it
- Label the equations (1) and (2).
- Multiply to match one coefficient.
- Add or subtract to eliminate it.
- Solve; substitute back; check.
Worked example
Solve 3x + 2y = 19 and 2x − 5y = −19.
- (1) × 5: 15x + 10y = 95. (2) × 2: 4x − 10y = −38.
- Add: 19x = 57, so x = 3.
- 3(3) + 2y = 19 → y = 5.
- Check (2): 6 − 25 = −19 ✓
x = 3, y = 5
2Substitution
- Best when one equation already says y = … or x = …
- Replace that letter in the other equation, solve, then substitute back.
Worked example
Solve y = 3x − 4 and 2x + y = 11.
- 2x + (3x − 4) = 11.
- 5x = 15, x = 3.
- y = 3(3) − 4 = 5.
x = 3, y = 5
3Forming simultaneous equations
- Choose a letter for each unknown and say what it means.
- Write one equation for each fact in the question.
- Answer in words with units at the end.
Worked example
2 adult tickets and 3 child tickets cost $31. 1 adult and 1 child ticket cost $12. Find the price of each.
- 2a + 3c = 31 and a + c = 12.
- From the second, a = 12 − c.
- 2(12 − c) + 3c = 31 → c = 7, a = 5.
Adult $5, child $7
4One linear and one non-linear equation
- Rearrange the linear equation to y = … (or x = …) and substitute into the non-linear one.
- This gives a quadratic: solve it to get two x values.
- Find the matching y for each x using the linear equation, and pair them.
- Graphically, the solutions are where the line meets the curve (0, 1 or 2 points).
Worked example
Solve y = x + 1 and x2 + y2 = 25.
- x2 + (x + 1)2 = 25 → 2x2 + 2x − 24 = 0.
- x2 + x − 12 = 0 → (x + 4)(x − 3) = 0.
- x = 3 → y = 4; x = −4 → y = −3.
(3, 4) and (−4, −3)
✗Common mistakes
| Don’t write | Write instead |
|---|---|
| Subtracting when the signs are different | Different signs: add the equations |
| −2y − (−2y) = −4y | −2y − (−2y) = 0 |
| Giving only the x values | Find each y and pair it with its x |
| Squaring (x + 1)2 as x2 + 1 | (x + 1)2 = x2 + 2x + 1 |
★In the exam
- Write what you do to each equation: (1) × 2, (2) × 3.
- Check your pair in the original equations — a quick way to catch sign errors.
- In word problems, state the answer in words with units.