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Chapter 10: Problem 14

Solve. $$\sqrt{y+4}+6=7$$

Short Answer

Expert verified
y = -3

Step by step solution

01

- Isolate the square root term

Subtract 6 from both sides of the equation to isolate the square root term. \(\sqrt{y+4}+6鈭�6=7鈭�6\)This simplifies to \(\sqrt{y+4} = 1\)
02

- Square both sides

Square both sides of the equation to eliminate the square root. \((\sqrt{y+4})^2 = 1^2\)This simplifies to \(y+4 = 1\)
03

- Solve for y

Subtract 4 from both sides of the equation to solve for y. \(y+4-4=1-4\)This simplifies to \(y=-3\)

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Key Concepts

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

Isolating the square root
To start solving the equation \( \sqrt{y+4} + 6 = 7 \), we first need to isolate the square root term. Isolating means getting the square root by itself on one side of the equation.

Here's how you can do it:

  • First, look at the equation and identify the square root term, which is \( \sqrt{y+4} \).
  • The term \( +6 \) is added to the square root. To isolate \( \sqrt{y+4} \), we need to remove the +6 by subtracting 6 from both sides of the equation.

    This is done like so:
    \( \sqrt{y+4} + 6 - 6 = 7 - 6 \).
  • Simplifying this, we get:
    \( \sqrt{y+4} = 1 \).


By removing the +6, we have successfully isolated the square root \( \sqrt{y+4} \). Now the equation is simpler and ready for the next step.
Squaring both sides of an equation
Once the square root is isolated, the next step is to eliminate it by squaring both sides of the equation. Squaring is simply raising both sides of the equation to the power of 2.

Let's see how this works:
  • Start with the isolated square root equation:
    \( \sqrt{y+4} = 1 \).
  • To remove the square root, you need to square both sides:
    \( (\sqrt{y+4})^2 = 1^2 \).
  • When you square a square root, the square and square root cancel each other out, leaving you with the term inside the square root:
    \( y+4 \).
  • So now the equation becomes:
    \( y+4 = 1 \).


Squaring both sides is a crucial step that helps simplify the equation further, making it easier to solve.
Solving for a variable
The final step involves solving for the variable, which in this case is \( y \). With the square root removed, we're left with a simpler equation that we can solve easily.

Here's what to do:
  • Start with the simplified equation from the previous step:
    \( y + 4 = 1 \).
  • To solve for \( y \), you need to isolate it. This means getting \( y \) by itself on one side of the equation.
  • Subtract 4 from both sides:
    \( y + 4 - 4 = 1 - 4 \).
  • Simplifying this, you get:
    \( y = -3 \).


Now, the variable \( y \) is isolated and its value is \( -3 \). So, the solution to the equation \( \sqrt{y+4} + 6 = 7 \) is \( y = -3 \). Don't forget to check your solution by plugging \( y = -3 \) back into the original equation to make sure it works!

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