Linear–quadratic simultaneous equations
Objective. Combine substitution with quadratic solution methods.
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- 2. Worked example
- 3. Practice
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Learn: the key idea
Linear–quadratic simultaneous equations are solved by substituting the linear equation into the quadratic one, producing a single quadratic equation in one variable. After rearranging the linear equation to make one variable the subject, substitute this expression into the quadratic equation wherever that variable appears, then solve the resulting quadratic using factorising or the formula. Each solution for one variable is then substituted back into the linear equation to find the matching value of the other variable, giving up to two pairs of solutions. Always check both solution pairs satisfy both original equations.
Worked example
Solve the simultaneous equations y = x + 1 and y = x² − 5 to find all pairs of solutions.
- 1Substitute y = x + 1 into the quadratic: x + 1 = x² − 5.
- 2Rearrange to form a quadratic: x² − x − 6 = 0.
- 3Factorise: (x − 3)(x + 2) = 0, giving x = 3 or x = −2.
- 4Find y using y = x + 1: y = 4 when x = 3, and y = −1 when x = −2.
Answer: (3, 4) and (−2, −1)
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Start practisingCommon mistake. Pupils substitute the quadratic equation into the linear one instead of the other way round, creating unnecessary complexity. Always make the linear equation the subject and substitute it into the quadratic.
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