Section 3.6 Systems of Linear Equations in Three Variables and Applications 283 1. Solutions to Systems of Linear Equations in Three Variables In Sections 3.1–3.3, we solved systems of linear equations in two variables. In this section, we will expand the discussion to solving systems involving three variables. A linear equation in three variables can be written in the form where A, B, and C are not all zero. For example, the equation Ax By Cz D, is a linear equation in three variables. Solutions to this equation 2x 3y z 6 are ordered triples of the form (x, y, z) that satisfy the equation. Some solutions to the equation 2x 3y z 6 are Solution: Check: ✔ True ✔ True ✔ True 11, 1, 12 2112 3112 112 6 12, 0, 22 2122 3102 122 6 10, 1, 32 2102 3112 132 6 Infinitely many ordered triples serve as solutions to the equation 2x 3y z 6. The set of all ordered triples that are solutions to a linear equation in three variables may be represented graphically by a plane in space. Figure 3-24 shows a portion of the plane 2x 3y z 6 in a 3-dimensional coordinate system. An example of a system of three linear equations in three variables is shown here. 2x y 3z 7 3x 2y z 11 2x 3y 2z 3 A solution to a system of linear equations in three variables is an ordered triple that satisfies each equation. Geometrically, a solution is a point of intersection of the planes represented by the equations in the system. A system of linear equations in three variables may have one unique solution, infinitely many solutions, or no solution (Table 3-2,Table 3-3, and Table 3-4). Table 3-2 One unique solution (planes intersect at one point) • The system is consistent. • The equations are independent. Concepts 1. Solutions to Systems of Linear Equations in Three Variables 2. Solving Systems of Linear Equations in Three Variables 3. Applications of Linear Equations in Three Variables 4. Solving Inconsistent Systems and Systems of Dependent Equations z x y Figure 3-24 Systems of Linear Equations in Three Variables and Applications Section 3.6
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