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SYSTEMS of EQUATIONS (Gaussian Elimination) A system of linear equations is a set of one or more linear equations. To solve a system of linear equations means to determine values that will simultaneously satisfy the unknowns in each equation. Ex Solve the following linear system using elimination: 2𝑥 + 𝑦 = −9 𝑥−𝑦 =6 Ex Determine the value(s) of a and b for which the given system has no solution, an infinite number of solutions, and one solution: 𝑥 + 4𝑦 = 𝑎 𝑥 + 𝑏𝑦 = 8 A linear system can be written without its variables in matrix form. A matrix is a rectangular array of numbers. Ex Express the given system in matrix form: 2𝑥 + 𝑦 − 𝑧 = 3 5𝑥 + 𝑧 = 2 2𝑦 + 4𝑧 = −3 2 1 −1 3 5 0 1 2 0 2 4 −3 AUGMENTED MATRIX The system can be solved using elementary row operations on the augmented matrix. This procedure is called row reduction. It produces a series of equivalent systems. ROW OPERATIONS: Any row can be multiplied or divided by a non–zero constant. Any row can be replaced by the sum or difference of that row and a multiple of another row. Any two rows can be interchanged. Ex Solve and classify each system: a) 𝑥 + 2𝑦 + 𝑧 = −4 𝑥 + 4𝑦 + 5𝑧 = −18 4𝑥 − 𝑧 = −4 b) 𝑥 + 2𝑦 + 3𝑧 = −4 𝑥 − 𝑦 − 3𝑧 = 8 2𝑥 + 𝑦 + 6𝑧 = −14 HOMEWORK: p.507–509 #3, 5, 6, 9, 12(omit d), 14, 15