What they are
- Simultaneous equations are two equations with two unknowns. The solution is the pair of values that makes both equations true.
- On a graph, the solution is the point where the two lines cross.
Elimination
- Add or subtract the equations to get rid of one letter. x + y = 10 and x − y = 4: adding gives 2x = 14, so x = 7 and y = 3.
- If no coefficients match, multiply first. 3x + 4y = 18 and 5x − 2y = 4: double the second to get 10x − 4y = 8, then add: 13x = 26, so x = 2 and y = 3.
- Same signs: subtract. Different signs: add. Always substitute back to find the other letter, then check in the other equation.
Substitution
- If one equation says y = …, put it into the other. y = 2x + 1 and 3x + y = 11: 3x + 2x + 1 = 11, so x = 2 and y = 5.
- (Higher) A linear and a quadratic equation: substitute the linear one into the quadratic. y = x + 1 and x² + y² = 13 give x² + (x + 1)² = 13, so x² + x − 6 = 0, which gives (2, 3) and (−3, −2).
Forming equations from words
- Choose a letter for each unknown, write two equations, then solve. '2 coffees and a tea cost £7; a coffee and a tea cost £4.20' gives 2c + t = 7 and c + t = 4.20, so c = £2.80.
Key terms
- Simultaneous equations
- Two equations with two unknowns that are true at the same time.
- Elimination
- Adding or subtracting equations to remove one unknown.
- Substitution
- Replacing a letter with an expression or value from another equation.
- Coefficient
- The number in front of a letter.
- Point of intersection
- The point where two graphs cross, which gives the solution.