Solve $\sqrt{y+4}-1=4$.
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)
Isolate the square root term by adding 1 to both sides: y + 4 = 5 .
Square both sides to eliminate the square root: y + 4 = 25 .
Solve for y by subtracting 4 from both sides: y = 21 .
Check the solution in the original equation to confirm validity: 21 .
Explanation
Problem Analysis We are given the equation y + 4 − 1 = 4 and asked to solve for y . Our goal is to isolate y by performing algebraic operations on both sides of the equation. First, we need to isolate the square root term. Then, we will square both sides to eliminate the square root and solve for y . Finally, we will check our solution to make sure it is not extraneous.
Isolating the Square Root To isolate the square root term, we add 1 to both sides of the equation: y + 4 − 1 + 1 = 4 + 1 y + 4 = 5
Eliminating the Square Root Now, we square both sides of the equation to eliminate the square root: ( y + 4 ) 2 = 5 2 y + 4 = 25
Solving for y To solve for y , we subtract 4 from both sides of the equation: y + 4 − 4 = 25 − 4 y = 21
Checking for Extraneous Solutions We must check our solution to ensure it is not extraneous. We substitute y = 21 back into the original equation: 21 + 4 − 1 = 25 − 1 = 5 − 1 = 4 Since the result is 4, which is equal to the right side of the original equation, our solution is valid.
Final Answer Therefore, the solution to the equation y + 4 − 1 = 4 is y = 21 .
Examples
Imagine you are designing a square garden and need to determine the length of each side. If the area of the garden plus a 4-foot wide path around it is 25 square feet, you can use the equation y + 4 = 5 to find the side length y of the garden. Solving this equation helps you determine the dimensions needed for your garden design, ensuring the garden fits within the specified area.
