Solve $\sqrt{2 y+9}=9$.
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The solution is $y=$ $\square$ .
(Simplify your answer. Type an integer or a fraction. Use a comma to separate answers as needed.)
B. There is no solution.
Answer (1)
Square both sides of the equation: ( 2 y + 9 ) 2 = 9 2 , which simplifies to 2 y + 9 = 81 .
Subtract 9 from both sides: 2 y = 72 .
Divide by 2: y = 36 .
The solution is 36 .
Explanation
Problem Analysis We are given the equation 2 y + 9 = 9 and asked to solve for y . Our goal is to isolate y on one side of the equation.
Squaring Both Sides To eliminate the square root, we square both sides of the equation: ( 2 y + 9 ) 2 = 9 2 This simplifies to: 2 y + 9 = 81
Isolating the Term with y Next, we subtract 9 from both sides of the equation to isolate the term with y : 2 y + 9 − 9 = 81 − 9 This simplifies to: 2 y = 72
Solving for y Now, we divide both sides of the equation by 2 to solve for y : 2 2 y = 2 72 This simplifies to: y = 36
Checking the Solution To check our solution, we substitute y = 36 back into the original equation: 2 ( 36 ) + 9 = 72 + 9 = 81 = 9 Since the left side equals the right side, our solution is correct.
Final Answer Therefore, the solution to the equation 2 y + 9 = 9 is y = 36 .
Examples
Imagine you're designing a square garden and need to determine the length of each side. If the area inside the garden, plus an extra 9 square feet for a small fountain, must equal 81 square feet, you can use the equation A + 9 = 9 to find the area A of the garden without the fountain. Solving this tells you how much space you have for planting, helping you plan your garden layout effectively. This kind of problem-solving is useful in many design and planning scenarios where you need to work backward from a known total to find an unknown component.
