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8.2 Systems of Linear Equations: Two Equations Containing Two Variables

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If each equation in a system of equations is linear, then we have a system of linear equations.

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If the graph of the lines in a system of two linear equations in two variables intersect, then the system of equations has one solution, given by the point of intersection. The system is consistent and the equations are independent. Solution y x

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If the graph of the lines in a system of two linear equations in two variables are parallel, then the system of equations has no solution, because the lines never intersect. The system is inconsistent. x y

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If the graph of the lines in a system of two linear equations in two variables are coincident, then the system of equations has infinitely many solutions, represented by the totality of points on the line. The system is consistent and dependent. x y

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Two Algebraic Methods for Solving a System 1. Method of substitution 2. Method of elimination

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STEP 1: Solve for x in (2) Use Method of Substitution to solve: (1) (2) add x and subtract 2 on both sides

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STEP 2: Substitute for x in (1) STEP 3: Solve for y

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STEP 4: Substitute y = -11/5 into result in Step1. Solution:

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STEP 5: Verify solution in

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Rules for Obtaining an Equivalent System of Equations 1. Interchange any two equations of the system. 2. Multiply (or divide) each side of an equation by the same nonzero constant. 3. Replace any equation in the system by the sum (or difference) of that equation and any other equation in the system.

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Multiply (2) by 2 Replace (2) by the sum of (1) and (2) Equation (2) has no solution. System is inconsistent. Use Method of Elimination to solve: (1) (2)

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Multiply (2) by 2 Replace (2) by the sum of (1) and (2) The original system is equivalent to a system containing one equation. The equations are dependent. Use Method of Elimination to solve: (1) (2)

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This means any values x and y, for which 2x -y =4 represent a solution of the system. Thus there is infinitely many solutions and they can be written as or equivalently

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