Simultaneous equations
Two equations with two unknowns have one pair of values that makes both true. Elimination and substitution are two routes to the same answer, and you can always check it at the end.
Part 1 of 3: Learn it
In short
- Elimination: make the coefficients of one letter match, then add or subtract the equations.
- Substitution: rearrange one equation for a letter and put it into the other.
- Always check your answer in the equation you did not use last.
Where this is in your specification
Spec points: DfE A19 and A21 (linear and quadratic pairs are Higher tier)
| Board | Topic: Simultaneous equations |
|---|---|
| DfE content | A19, A21 |
| AQA 8300 | Algebra |
| Edexcel 1MA1 | Algebra |
| OCR J560 | 6 Algebra |
Elimination
- Line up the x terms, y terms and numbers.
- Multiply one or both equations so one letter has the same coefficient in both.
- Same signs: subtract the equations. Different signs: add them.
- Solve for the remaining letter, then substitute back to find the other.
Substitution
If one equation already says y = ... (or is easy to rearrange), replace y in the other equation with that expression. This is the method for a linear and a quadratic equation together (Higher), such as y = x + 1 and x² + y² = 25.
Graphs
Each linear equation is a straight line. The solution is the point where the lines cross. A graphical answer is only as accurate as the drawing, so algebra is better when exact answers are needed. Parallel lines never cross, so their equations have no solution.
Forming equations
Choose letters for the two unknowns, write one equation for each fact in the question, then solve. For example: 3 coffees and 2 teas cost £8.70, and 1 coffee and 2 teas cost £4.50. With c and t in pounds: 3c + 2t = 8.70 and c + 2t = 4.50.
Solve x + 2y = 11 and x − y = 2.
Show the answer
Taking one equation away from the other leaves 3y = 9. So y = 3, and then x = 5.
Part 2 of 3: See it worked
Worked examples
Example 1
Solve 2x + 3y = 13 and 5x − 2y = 4.
- Multiply the first by 2 and the second by 3: 4x + 6y = 26 and 15x − 6y = 12
- Different signs on 6y, so add: 19x = 38, x = 2
- Substitute: 2(2) + 3y = 13, so 3y = 9, y = 3
- Check in the second: 5(2) − 2(3) = 10 − 6 = 4
Answer: x = 2, y = 3.
Example 2
3 coffees and 2 teas cost £8.70; 1 coffee and 2 teas cost £4.50. Find the price of each.
- 3c + 2t = 8.70 and c + 2t = 4.50
- Same sign on 2t, so subtract: 2c = 4.20, c = 2.10
- 2.10 + 2t = 4.50, so t = 1.20
- Check: 3(2.10) + 2(1.20) = 6.30 + 2.40 = 8.70
Answer: A coffee costs £2.10 and a tea costs £1.20.
Common mistakes
- Multiplying only one term of an equation instead of every term.
- Subtracting when the signs are different, which doubles the letter instead of removing it.
- Sign slips when subtracting a negative: 4 − (−6) = 10.
- Stopping after finding one letter.
Solve y = 2x and 3x + y = 15.
Show the answer
3x + 2x = 15, so x = 3 and y = 6.
Part 3 of 3: Test yourself
Check yourself
Answer each one in your head or on paper first, then open it to check.
Solve x + 2y = 11 and x − y = 2.
Taking one equation away from the other leaves 3y = 9. So y = 3, and then x = 5.
Solve y = 2x and 3x + y = 15.
3x + 2x = 15, so x = 3 and y = 6.
Two lines have the same gradient and different intercepts. How many solutions?
None: the lines are parallel and never meet.
Jobs that use this
Each link opens the job profile on the National Careers Service (England). In the rest of the UK: My World of Work (Scotland), Careers Wales, nidirect careers (Northern Ireland).
Estos apuntes están en inglés porque siguen los programas de examen del Reino Unido.
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