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Published byAudra Carpenter Modified over 7 years ago

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SOLUTION EXAMPLE 1 A linear system with no solution Show that the linear system has no solution. 3x + 2y = 10 Equation 1 3x + 2y = 2 Equation 2 Graph the linear system. METHOD 1 Graphing

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EXAMPLE 1 A linear system with no solution ANSWER The lines are parallel because they have the same slope but different y- intercepts. Parallel lines do not intersect, so the system has no solution.

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EXAMPLE 1 A linear system with no solution ANSWER The variables are eliminated and you are left with a false statement regardless of the values of x and y. This tells you that the system has no solution. Subtract the equations. METHOD 2 Elimination 3x + 2y = 10 3x + 2y = 2 0 = 8 This is a false statement.

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EXAMPLE 2 A linear system with infinitely many solutions Show that the linear system has infinitely many solutions. x – 2y = –4 Equation 1 Equation 2 y = x + 2 1 2 SOLUTION Graphing METHOD 1 Graph the linear system.

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EXAMPLE 2 A linear system with infinitely many solutions ANSWER The equations represent the same line, so any point on the line is a solution. So, the linear system has infinitely many solutions.

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EXAMPLE 2 A linear system with infinitely many solutions Substitute x + 2 for y in Equation 1 and solve for x. 1 2 x – 2y = –4 Write Equation 1. 2 Substitute x + 2 for y. 1 x – 2 x + 2 = 1 2 –4 METHOD 2 Substitution The variables are eliminated and you are left with a statement that is true regardless of the values of x and y. This tells you that the system has infinitely many solutions. ANSWER –4 = –4 Simplify.

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GUIDED PRACTICE for Examples 1 and 2 1. 5x + 3y = 6 –5x – 3y = 3 Tell whether the linear system has no solution or infinitely many solutions. Explain. ANSWER No solution. Sample answer: When you solve the system you get 0 = 9, which is a false statement.

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GUIDED PRACTICE for Examples 1 and 2 2. y = 2x – 4 –6x + 3y = –12 Tell whether the linear system has no solution or infinitely many solutions. Explain. ANSWER Infinitely many solutions. Sample answer: When you solve the system you get 12 = 12, which is a true statement.

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