Simultaneous equations
Two equations, two unknowns: solve by elimination or substitution. Higher tier adds linear+quadratic.
Foundation: two linear equations. Higher: one linear, one quadratic. The pair has 0, 1, 2 or infinitely many solutions depending on geometry.
Worked examples
3x + 2y = 16 and x − y = 2. Sub y = x − 2 ⇒ 3x + 2(x−2) = 16 ⇒ 5x = 20 ⇒ x = 4, y = 2.
Elimination: multiply equations to align a coefficient, then add/subtract to remove a variable.
Higher: y = x + 1 and x² + y² = 25. Sub: x² + (x+1)² = 25 ⇒ 2x² + 2x − 24 = 0 ⇒ x = 3 or −4.
Frequently asked questions
Elimination or substitution?
Elimination is cleaner when coefficients align easily. Substitution is essential for linear+quadratic.
Graphical method?
Plot both equations; intersections are solutions. Good visual check.