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Linear and Quadratic Simultaneous Equations Calculator

Solve a linear equation and a quadratic together, the GCSE Higher way: substitute the line into the circle or curve, tidy it into a quadratic equal to zero, solve that, then use the line to find each y. A graph shows the line crossing the curve, so you can see why there are two answers.

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    Worked Example

    Solve y = x + 1 and x² + y² = 25.

    1. Substitute:x² + (x + 1)² = 25
    2. Expand and tidy:2x² + 2x − 24 = 0, so x² + x − 12 = 0
    3. Factorise:(x + 4)(x − 3) = 0, so x = −4 or x = 3
    4. Use the line:x = 3 gives y = 4, x = −4 gives y = −3

    Answer: x = 3, y = 4 and x = −4, y = −3

    Try These Yourself

    Solve y = x + 2 and y = x² − 4.Show answerHide answer

    Answer: x = −2, y = 0 and x = 3, y = 5

    x² − 4 = x + 2 gives x² − x − 6 = 0, which is (x − 3)(x + 2) = 0.

    Solve y = 2x and x² + y² = 20.Show answerHide answer

    Answer: x = −2, y = −4 and x = 2, y = 4

    x² + 4x² = 20, so 5x² = 20 and x² = 4.

    Does y = 10 meet x² + y² = 25?Show answerHide answer

    Answer: No, the line misses the circle

    x² + 100 = 25 gives x² = −75, which has no real solution.