308 Chapter 3 Systems of Linear Equations and Inequalities Systems of Linear Equations in Three Variables and Applications Key Concepts A linear equation in three variables can be written in the form Ax By Cz D, where A, B, and C are not all zero.The graph of a linear equation in three variables is a plane in space. A solution to a system of linear equations in three variables is an ordered triple that satisfies each equation. Graphically, a solution is a point of intersection among three planes. A system of linear equations in three variables may have one unique solution, infinitely many solutions (dependent equations), or no solution (inconsistent system). Examples Example 1 A x 2y z 4 B 3x y z 5 C 2x 3y 2z 7 A x 2y z 4 B 3x y z 5 4x y 9 D 2 A 2x 4y 2z 8 C 2x 3y 2z 7 D 4x y 9 4x y 9 E 4x 7y 15 4x 7y 15 6y 6 y 1 D E Substitute into either equation or . 4x 112 D 9 4x 8 x 2 x 2 y 1 A B Substitute and into equation , , or C . A 122 2112 z 4 z 0 The solution set is 512, 1, 026. y 1 4x 7y 15 E Section 3.6 Multiply by 1.
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