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7.1 - Systems of Linear Equations What is the solution of a line?? Solution of 2x + y = 5? y 8 7 6 5 4 3 2 1 -8 -7 -6 -5 -4 -3 -2 -1 0 -1 -2 -3 -4 -5 -6 -7 -8 1 2 3 4 5 6 7 8 x System of Linear Equations: 2x + 3y = -1 -x + y = -3 OR 2x - 3y - z = 0 -x + 2y + z = 5 3x - 4y - z = 1 System of Equations - 2 or more equations, each containing 1 or more variables Is (2, 1 ) a solution for: 2x + y = 5 -4x + 6y = -2 Check: Solution of a system - x, y (& z for 3 equations) values that satisfy each equation of the system To solve a system, must find all such solutions System of linear equations is 2 (or 3) lines Lines either: 1 ) Intersect 2) Parallel 3) Coincident (same line) Possible solutions to a system: 1 ) one solution (intersect) 2) No solution (parallel) 3) Infinite # of solutions (coincident) Consistent system - system that has at least one solution Inconsistent system - system that has no solution Dependent system - system where any solution of one equation is a solution of the other Independent system - system where solutions of one equation is not necessarily a solution of the other equation 3 ways to solve: 1 ) Graphing -graph each line and find point of intersection 2) Substitution -solve for 1 variable in terms of the other in 1 equation -substitute the result in the remaining equation and solve -find value of remaining variable by back substitution 3) Elimination or Addition -multiply each side of an equation(s) by the same nonzero constant -add the 2 equations to eliminate one of the variables -solve for remaining variable -use back substitution to solve for variable that was eliminated Graphing Ex: Find each solution by graphing. State if the system is consistent or inconsistent, and dependent or independent. 1 . 3x + 2y = 2 x - 7y = -30 y 10 9 8 7 6 2. x - y = 3 -3x + y = 1 5 4 3 2 1 -1 0 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 -1 -2 -3 -4 -5 -6 -7 -8 -9 -1 0 1 2 3 4 5 6 7 8 9 10 x Substitution Method Ex: Find each solution using substitution. State if the system is consistent or inconsistent, and dependent or independent. 1 . x + 3y = 5 2x - 3y = -8 2. x - y = 5 -3x + 3y = 2 Elimination Method Ex: Find solution using elimination. State if the system is consistent or inconsistent, and dependent or independent. 1 . 3x + 3y = -1 4x + y = 8 3 2. 2x + 4y = 2 3 3x - 5y = -1 0