Problem 17 - Elementary Algebra

Question

For the following problems, solve the square root equations. $$ \sqrt{y-4}-4=0 $$

Step-by-Step Solution

Answer

Answer: The value of y in the given equation is 20.

Step 1: Isolate the square root term

To start, we want to isolate the square root term on one side of the equation. To do this, we'll add 4 to both sides of the equation: $$ \sqrt{y-4} = 4 $$

Step 2: Square both sides of the equation

Now that we've isolated the square root term, we'll square both sides of the equation to eliminate the square root. This will give us a linear equation that we can solve for y: $$ (\sqrt{y-4})^2 = 4^2 $$ Simplifying this equation, we get: $$ y-4 = 16 $$

Step 3: Solve for y

Finally, we'll solve for y by adding 4 to both sides of the equation: $$ y = 16 + 4 $$ So, the solution to the given equation is: $$ y = 20 $$

Key Concepts

Isolating the Square Root Term

When dealing with square root equations, the first crucial step is isolating the square root term. This means rearranging your equation so that the square root is by itself on one side.
This process helps to simplify the equation and makes it easier to handle in the subsequent steps. For the equation \(\sqrt{y-4} - 4 = 0\), we can isolate the square root term by adding 4 to both sides.

• Adding 4 counteracts the \(-4\) on the left side.
• This transforms our equation into \(\sqrt{y-4} = 4\).

With the square root term now isolated, you have set the stage for the next step in solving the equation.

Squaring Both Sides

Once the square root term is isolated, the next step involves eliminating the square root by squaring both sides of the equation.
This technique converts the equation from one involving a square root to a more familiar linear equation. In our example, we take \(\sqrt{y-4} = 4\) and square each side:

• Squaring the left side: \((\sqrt{y-4})^2\).
• Squaring the right side: \(4^2\).

Squaring the left side effectively removes the square root, transforming \(\sqrt{y-4}\) into \(y-4\). Similarly, \(4^2\) becomes 16.
The resulting linear equation is \(y-4 = 16\), making the process of finding the variable straightforward.

Solving Linear Equations

After eliminating the square root, what remains is a simple linear equation to solve.
In the example, the new equation is \(y-4 = 16\). To solve for \(y\), you need to isolate it on one side:

• Add 4 to both sides of the equation: \(y - 4 + 4 = 16 + 4\).
• Simplifying gives \(y = 20\).

This step involves basic arithmetic, making it less complex compared to the earlier parts of the problem.
Once you perform these calculations, you have reached the solution: \(y = 20\). Understanding each step and performing operations with care ensures that you solve square root equations correctly.