Simultaneous Equations

Simultaneous equations are two (or more) equations that must be satisfied at the same time. The solution is the set of values that works in both equations. This lesson covers solving by elimination, substitution, and linear–quadratic simultaneous equations (Higher).

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Key Vocabulary

Term	Meaning
Simultaneous equations	A set of equations with the same unknowns, solved together
Elimination	Adding or subtracting equations to remove one variable
Substitution	Replacing one variable with an expression from the other equation
Linear equation	An equation where the highest power of the variable is 1
Linear–quadratic pair	One linear and one quadratic equation solved simultaneously

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Solving by Elimination

The idea: make the coefficients of one variable the same in both equations, then add or subtract to eliminate it.

Worked Example 1

Solve: 2x + 3y = 12 ... (1) 5x + 3y = 21 ... (2)

The y-coefficients are already the same (both 3y).

Subtract (1) from (2): (5x + 3y) − (2x + 3y) = 21 − 12

3x = 9 → x = 3