Algebra and Trigonometry, 5th Edition

9.2 Systems of Equations in Three Variables

• Solve systems of linear equations in three variables.

• Use systems of three equations to solve applied problems.

• Model a situation using a quadratic function.

A linear equation in three variables is an equation equivalent to one of the form $Ax+By+Cz=D$, where A, B, C, and D are real numbers and none of A, B, and C is 0. A solution of a system of three equations in three variables is an ordered triple that makes all three equations true. For example, the triple $(\mathrm{2,}-\mathrm{1,}0)$ is a solution of the system of equations

$$\begin{array}{ccccccc}4x& +& 2y& +& 5z& =& 6,\\ 2x& -& y& +& z& =& 5,\\ 3x& +& 2y& -& z& =& 4.\end{array}$$

We can verify this by substituting 2 for x, $-1$ for y, and 0 for z in each equation.

Solving Systems of Equations in Three Variables

We will solve systems ...