EXAMPLE 4 Solve linear systems with many or no solutions Solve the linear system. a.x – 2y = 4 3x – 6y = 8 b.4x – 10y = 8 – 14x + 35y = – 28 SOLUTION a.

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EXAMPLE 4 Solve linear systems with many or no solutions Solve the linear system. a.x – 2y = 4 3x – 6y = 8 b.4x – 10y = 8 – 14x + 35y = – 28 SOLUTION a. Because the coefficient of x in the first equation is 1, use the substitution method. Solve the first equation for x. x – 2y = 4 x = 2y + 4 Write first equation. Solve for x.

EXAMPLE 4 Solve linear systems with many or no solutions Substitute the expression for x into the second equation. 3x – 6y = 8 3(2y + 4) – 6y = 8 12 = 8 Write second equation. Substitute 2y + 4 for x. Simplify. Because the statement 12 = 8 is never true, there is no solution. ANSWER

EXAMPLE 4 Solve linear systems with many or no solutions b. Because no coefficient is 1 or – 1, use the elimination method. Multiply the first equation by 7 and the second equation by 2. 4x – 10y = 8 – 14x + 35y = – 28 28x – 70y = 56 – 28x + 70y = – 56 Add the revised equations. 0 = 0 ANSWER Because the equation 0 = 0 is always true, there are infinitely many solutions.

GUIDED PRACTICE for Example 4 Solve the linear system using any algebraic method. 5.12x – 3y = – 9 –4x + y = 3 infinitely many solutions ANSWER 6.6x + 15y = – 12 – 2x – 5y = 9 ANSWER no solutions

GUIDED PRACTICE for Example 4 Solve the linear system using any algebraic method. 8.12x – 2y = 21 3x + 12y = – 4 The solution is (, ) ANSWER – x + 3y = 20 – x – y = – infinitely many solutions ANSWER

GUIDED PRACTICE for Example 4 Solve the linear system using any algebraic method. 9.8x + 9y = 15 5x – 2y = 17 5x + 5y = x + 3y = 4.2 (3, –1) ANSWER (0.6, 0.4) ANSWER