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SOLUTION

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

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SOLUTION

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  1. 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 SOLUTION METHOD 1 Graphing Graph the linear system.

  2. 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. EXAMPLE 1 A linear system with no solution

  3. 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. 3x + 2y = 10 3x + 2y = 2 0 = 8 This is a false statement. EXAMPLE 1 A linear system with no solution METHOD 2 Elimination Subtract the equations.

  4. Graphing METHOD 1 y = x + 2 1 2 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 SOLUTION Graph the linear system.

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

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

  7. ANSWER No solution. Sample answer: When you solve the system you get 0 = 9, which is a false statement. for Examples 1 and 2 GUIDED PRACTICE Tell whether the linear system has no solution or infinitely many solutions. Explain. 1.5x + 3y = 6 –5x – 3y = 3

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

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