Solving a System with Three Unknowns and Two Equations

Example

Solve the system:

$x+2y+z=4$

$3x-y-4y=-9$

Solution

Geometrically, the solution is the intersection of the two planes given by the two equations above. The intersection of two different nonparallel planes is a line. Thus there will be an infinite number of ordered triples in the solution set, representing the points on the line of intersection.

To eliminate x, multiply both sides of the first equation by −3 and add the result to the second equation. (Either y or z could have been eliminated instead.)

$-3x-6y-3z=-12$

$3x-y-4z=-9$

___________________________

$-7y-7z=-21$

$-7z=7y-21$

$z=-y+3$

This gives z in terms of y. We also need to express x in terms of y by solving the first original equation for x and substituting −y + 3 for z in the result.

$x+2y+z=4$

$x=-2y-z+4$

$x=-2y-\left(-y+3\right)+4$

$x=-y+1$

The system has an infinite number of solutions. With y arbitrary, the solution set is of the form {(−y + 1, y, −y + 3)}.

Note

Had we solved here for y instead of z, the solution would have had a different form but would have led to the same set of solutions. In that case we would have z arbitrary, and the solution set would be of the form {(z − 2, −z + 3, z)}. By choosing z = 2, one solution would be (0, 1, 2).