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SOLVING SYSTEMS OF LINEAR EQUATIONS AND INEQUALITIES

A SYSTEM OF EQUATIONS is a set of equations with the same variables. EXAMPLE 2x + 3y = 6 x – 2y = 7 What this chapter is about is finding the solution(s) for these systems. In the first section (7.1), we will look at solving systems by graphing.

POSSIBILITIESPOSSIBILITIES LINES INTERSECT AT ONE POINT ONE SOLUTION – where the lines cross (written as an ordered pair – see blue dot) THE SYSTEM IS CONSISTENT & INDEPENDENT LINES ARE PARALLEL NO SOLUTION (The lines do not cross) THE SYSTEM IS INCONSISTENT. LINES COINCIDE (same line) INFINITE SOLUTIONS (infinite number of points in common) THE SYSTEM IS CONSISTENT & DEPENDENT.

DEFINITIONS CONSISTENT A SYSTEM OF EQUATIONS THAT HAS AT LEAST ONE ORDERED PAIR THAT SATISFIES BOTH EQUATIONS. INCONSISTENT A SYSTEM OF EQUATIONS WITH NO ORDERED PAIR THAT SATISFIES BOTH EQUATIONS INDEPENDENT DEPENDENT A SYSTEM OF EQUATIONS WITH EXACTLY ONE SOLUTION. A SYSTEM OF EQUATIONS THAT HAS AN INFINITE NUMBER OF SOLUTIONS.

RECAPRECAP IF LINES CROSS: There is one solution. The system is consistent and independent. IF LINES ARE PARALLEL: There Is no solution. The system is inconsistent. IF LINES COINCIDE (same line): There are an infinite number of solutions. The system is consistent and dependent. Slopes are different. Slopes are the same, y-intercepts are different. Slopes are the same, y-intercepts are same.

PRACTICE SOLVE BY GRAPHING 1. y = x + 3 y = -x -1 2. 2x + 3y = 6 2x + 3y = 2 3. 2x - y = 4 y = 2x - 4 PROBLEM #1 SOLUTION: (-2, 1) 1. y = x + 3 y = -x -1

PRACTICE SOLVE BY GRAPHING 1. y = x + 3 y = -x -1 2. 2x + 3y = 6 2x + 3y = 2 3. 2x - y = 4 y = 2x - 4 PROBLEM #2 NO SOLUTION

PRACTICE SOLVE BY GRAPHING 1. y = x + 3 y = -x -1 2. 2x + 3y = 6 2x + 3y = 2 3. 2x - y = 4 y = 2x - 4 PROBLEM #3 INFINITE SOLUTIONS

YOU CAN FIND THE EXACT SOLUTION OF A SYSTEM BY USING SUBSTITUTION. SOLVE: y = 2x 2x + 5y = 12 STEP 1: Solve one equation for a variable. In this case, the 1 st equation is already solved for y. STEP 2: Substitute into 2 nd equation and solve. 2x + 5y = 12 2x + 5(2x) = 12 2x + 10x = 12 12x = 12 So, x = 1

YOU CAN FIND THE EXACT SOLUTION OF A SYSTEM BY USING SUBSTITUTION. SOLVE: y = 2x 2x + 5y = 12 STEP 3: Substitute value in one of the equations. y = 2x (1, 2) y = 2(1) y = 2 You can check your answers

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