Three simultaneous linear equations

Choose a variable whose coefficients make elimination manageable. Combine the first equation with the second, then the first with the third, eliminating the same variable each time. These two resulting equations can be solved by the ordinary two-variable method. Record which equations were combined so that signs can be checked.

Substitute the two values into an original equation to recover the third, then check all three original equations. A contradiction means there is no solution. A zero equation may mean the equations are dependent; it does not automatically imply a unique solution. The method does not require a matrix inverse.

Worked example

Solve x + y + z = 6, x − y + z = 2, and 2x + y − z = 1.

1. Subtract the second equation from the first: 2y = 4, so y = 2.
2. The first now gives x + z = 4; the third gives 2x − z = −1.
3. Add to obtain 3x = 3, then find z.

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

Revise first: Simultaneous linear and quadratic equations.