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3.1 Solving Linear Systems by Graphing

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System of two linear equations: two linear equationsin two variables x and y consists of equations on the following form:Ax + By = C Equation 1Dx + Ey = F Equation 2

Solution: a solution of a system of linear equations in two variables is an ordered pair (x, y) that satisfies both equations.

Checking a Solution: In order to check to see if an ordered pair is a solution for the system you must plug in the ordered pair to both

equations. Then it is a solution if the ordered pair works for both of the equations.Example 1: Check whether (a) (1,4) and (b) (-5, 0) are solutions of the following system.

x – 3y = –5 –2x + 3y = 10

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Steps for Solving a Linear System using Graphing

Write each equation in a form that is easy to graph.

Graph both equations in the same coordinate plane.

Estimate the coordinates of the point of intersection.

Check the coordinates algebraically by substituting into each equation of the original linear system.

Example 2: Solve the systems by the graphing method.a. 2x – 2y = –8 2x + 2y = 4

b. –4x – 3y = –12 4x + 2y = 8

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Number of solutions of a linear system- The relationship between the graph of a linear system and the system’s number of solutions is described below.

GRAPHICAL ALGEBRAICINTERPRETATION INTERPRETATION

The graph of the system The system has exactly one is a pair of lines that intersect solutionin one point

The graph of the system is a The system has infinitely single line many solutions

The graph of the system is a The system has no solutionpair of parallel lines so that there is no point of intersection.

 

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Exactly one Infinitely many solutions No solutionsolution

Examples: Tell how many solutions the system has.1. 2x + 4y = 12 x + 2y = 62. x – y = –15 2x – 2y = 9