How do you solve #3sqrt(y+3) = 3sqrt(2y-7)# and find any extraneous solutions?

Question

Isaiah Anthony · Accepted Answer

y=10

square the two edges, #9(y+3)=9(2y-7)# #9y+27=18y-63# #18y-9y=27+63# #9y=90# #y=10#

Jameson Allen · Answer

undefined To solve the equation 3√(y+3) = 3√(2y-7) and find any extraneous solutions, we can follow these steps:

1. Start by cubing both sides of the equation to eliminate the square roots:
   (3√(y+3))^3 = (3√(2y-7))^3

2. Simplify each side by cubing the expressions:
   27(y+3) = 27(2y-7)

3. Distribute the 27 on both sides of the equation:
   27y + 81 = 54y - 189

4. Combine like terms by moving all the y terms to one side and the constant terms to the other side:
   27y - 54y = -189 - 81

5. Simplify the equation:
   -27y = -270

6. Divide both sides of the equation by -27 to solve for y:
   y = 10

7. Check for extraneous solutions by substituting the found value of y back into the original equation:
   3√(10+3) = 3√(2(10)-7)
   3√13 = 3√13

8. Since the equation holds true, there are no extraneous solutions.

Therefore, the solution to the equation 3√(y+3) = 3√(2y-7) is y = 10, and there are no extraneous solutions.