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Chapter 3: Problem 48

Solve. $$\sqrt{y-5}-2=3$$

Short Answer

Expert verified
y = 30

Step by step solution

01

Isolate the square root term

Start by adding 2 to both sides of the equation to isolate the square root term. \[\sqrt{y-5} - 2 + 2 = 3 + 2\]This simplifies to:\[\sqrt{y-5} = 5\]
02

Square both sides

Next, square both sides to eliminate the square root. \[(\sqrt{y-5})^2 = 5^2\]This simplifies to:\[y-5 = 25\]
03

Solve for y

Finally, add 5 to both sides to solve for y. \[y-5 + 5 = 25 + 5\]This simplifies to:\[y = 30\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Isolating the Square Root
When solving square root equations, the first step is to isolate the square root term. This means you want the square root to be by itself on one side of the equation. In the exercise \(\sqrt{y-5} - 2 = 3\), follow these steps to isolate the square root:
  • Add 2 to both sides of the equation: \(\sqrt{y-5} - 2 + 2 = 3 + 2\)
  • This results in: \(\sqrt{y-5} = 5\)
By performing this operation, you have successfully isolated the square root term. This is crucial to make the equation easier to deal with and to eventually solve for the variable inside the square root.
Squaring Both Sides
Once you've isolated the square root, the next step is to eliminate it by squaring both sides of the equation. This step removes the square root sign, making it easier to solve the equation. In the equation \(\sqrt{y-5} = 5\), we square both sides to get rid of the square root:
  • Square the left side: \(\left( \sqrt{y-5} \right)^2 = y-5\)
  • Square the right side: \(5^2 = 25\)
  • This gives us: \(y-5 = 25\)
By squaring both sides, we are left with a simple linear equation that is easier to solve.
Solving Linear Equations
The final step in solving the problem is to solve the resulting linear equation. After squaring both sides, you are left with \(y-5 = 25\). To solve for \y\, follow these steps:
  • Add 5 to both sides: \(y-5 + 5 = 25 + 5\)
  • This simplifies to: \(y = 30\)
Solving linear equations generally involves isolating the variable. In this case, adding 5 to both sides isolates \y\ and gives you the final answer. This concludes the solution, where \y = 30\.
Just remember: solving square root equations often involves isolating the square root, squaring both sides, and then solving a simpler linear equation.

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