 # Identify the number of solutions of an equation

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Identify the number of solutions of an equation
EXAMPLE 4 Identify the number of solutions of an equation Solve the equation, if possible. a. 3x = 3(x + 4) b. 2x + 10 = 2(x + 5) SOLUTION a. 3x = 3(x + 4) Original equation 3x = 3x + 12 Distributive property The equation 3x = 3x + 12 is not true because the number 3x cannot be equal to 12 more than itself. So, the equation has no solution. This can be demonstrated by continuing to solve the equation.

Identify the number of solutions of an equation
EXAMPLE 4 Identify the number of solutions of an equation 3x – 3x = 3x + 12 – 3x Subtract 3x from each side. 0 = 12 Simplify. ANSWER The statement 0 = 12 is not true, so the equation has no solution.

Identify the number of solutions of an equation
EXAMPLE 1 EXAMPLE 4 Identify the number of solutions of an equation b. 2x + 10 = 2(x + 5) Original equation 2x + 10 = 2x + 10 Distributive property ANSWER Notice that the statement 2x + 10 = 2x + 10 is true for all values of x.So, the equation is an identity, and the solution is all real numbers.

GUIDED PRACTICE for Example 4 8. 9z + 12 = 9(z + 3) SOLUTION
Original equation 9z + 12 = 9z + 27 Distributive property The equation 9z + 12 = 9z + 27 is not true because the number 9z + 12 cannot be equal to 27 more than itself. So, the equation has no solution. This can be demonstrated by continuing to solve the equation.

The statement 12 = 27 is not true, so the equation has no solution.
GUIDED PRACTICE for Example 4 9z – 9z + 12 = 9z – 9z + 27 Subtract 9z from each side. 12 = 27 Simplify. ANSWER The statement 12 = 27 is not true, so the equation has no solution.

GUIDED PRACTICE for Example 4 9. 7w + 1 = 8w + 1 SOLUTION – w + 1 = 1
Subtract 8w from each side. – w = 0 Subtract 1 from each side. ANSWER w = 0

GUIDED PRACTICE for Example 4 10. 3(2a + 2) = 2(3a + 3) SOLUTION
Original equation 6a + 6 = 6a + 6 Distributive property ANSWER The statement 6a + 6 = 6a + 6 is true for all values of a. So, the equation is an identity, and the solution is all real numbers.