Simultaneous equations

Description

Simultaneous equations are two (or more) equations with the same two unknowns. You solve them together to find values of both variables.

Method 1 — Elimination: multiply equations to make the coefficients of one variable the same, then add or subtract. $$2x + 3y = 11 \quad (1)$$ $$4x - 3y = 7 \quad (2)$$ Adding: $6x = 18$, so $x = 3$. Substitute: $y = \frac{11 - 6}{3} = \frac{5}{3}$.

Method 2 — Substitution: make one variable the subject of one equation and substitute into the other.

Solving graphically: plot both lines; the solution is the point of intersection.

Higher — one linear, one quadratic: substitute the linear into the quadratic and solve the resulting quadratic.

Common error: having the signs wrong when subtracting equations — watch out for negative coefficients.

Links

Manipulating algebraic fractions Solving linear equations