Explore: Solutions to Systems of Equations with Three Variables Solutions

Systems of equations with three variables can have one solution, no solutions, or infinite solutions.
For systems with one solution, the answer is often written as an ordered triple , (x, y, z).
For systems with no solution, there is no common intersection point among the three equations.
For systems with an infinite number of solutions, a solution of 0 = 0 occurs for a combination of two or more equations.

Determine whether the ordered triple (22.5, –12, 0.5) is a solution to the system of equations.

System A

$x + y - z = 10 3 x - 2 y + z = 0 x - y + z = 11$

Implement

$(22.5) + (- 12) - (0.5) = 10 ✓ 3 (22.5) - 2 (- 12) + (0.5) = 92 ✖ (22.5) - (- 12) + (0.5) = 35 ✖$

Explain

The ordered triple did not make all equations true. Move on to the next system.

System B

$x + 2 y + z = - 1 x + 2 y + 5 z = 1 2 x + 3 y + 4 z = 11$

Implement

$(22.5) + 2 (- 12) + (0.5) = - 1 ✓ (22.5) + 2 (- 12) + 5 (0.5) = 1 ✓ 2 (22.5) + 3 (- 12) + 4 (0.5) = 11 ✓$

Explain

When the ordered triple is substituted into the equations, they are all true. Therefore this is the solution to system B.