Free Maths Calculator

Three Unknowns Simultaneous Equations Calculator

Solve three simultaneous equations in x, y and z by elimination. The calculator removes z from two different pairs, which leaves an ordinary pair in x and y, solves that, then substitutes back up the chain. Every scaled equation is labelled, so each line can be matched to your own working.

Try It: Enter Your Numbers

Your question

Updates as you type

Answer

Method, Step by Step

    Worked Example

    Solve x + y + z = 6, 2x − y + z = 3 and x + 2y − z = 2.

    1. Subtract to remove z, (2) − (1):x − 2y = −3, call this (4)
    2. Add to remove z, (3) + (1):2x + 3y = 8, call this (5)
    3. From (4) × 2 − (5):−7y = −14, so y = 2, then x = 1
    4. Substitute into (1):1 + 2 + z = 6, so z = 3

    Answer: x = 1, y = 2, z = 3

    Try These Yourself

    Solve x + y + z = 9, x − y = 1 and y − z = 1.Show answerHide answer

    Answer: x = 4, y = 3, z = 2

    Write x = y + 1 and z = y − 1, then (y + 1) + y + (y − 1) = 9, so 3y = 9.

    Solve x + y + z = 3, x + 2y + 3z = 6 and x − y + z = 1.Show answerHide answer

    Answer: x = 1, y = 1, z = 1

    Subtract the first from the third: −2y = −2, so y = 1.