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Chapter 7: Problem 23

Solve. $$ 2 y^{1 / 2}-13=7 $$

Short Answer

Expert verified
y = 100

Step by step solution

01

Isolate the Square Root Term

First, add 13 to both sides of the equation to isolate the term with the square root. \[ 2y^{1/2} - 13 + 13 = 7 + 13 \]Which simplifies to:\[ 2y^{1/2} = 20 \]
02

Divide by 2

Next, divide both sides by 2 to solve for the square root of y.\[ \frac{2y^{1/2}}{2} = \frac{20}{2} \]This gives us:\[ y^{1/2} = 10 \]
03

Square Both Sides

To find y, square both sides of the equation:\[ (y^{1/2})^2 = 10^2 \]Which simplifies to:\[ y = 100 \]

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Key Concepts

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

Isolate the Square Root Term
In order to solve an equation involving a square root, the first step is to isolate the square root term. This means we want to have the term with the square root on one side of the equation and everything else on the opposite side.

For our problem, the equation is given as: If we start with 2y^{1/2} - 13 = 7.

We want to move the -13 to the other side of the equation. We can do this by adding 13 to both sides: 2y^{1/2} - 13 + 13 = 7 + 13.
Which simplifies to: 2y^{1/2} = 20.
Notice now that the term with the square root is by itself on one side of the equation.

Operations with Square Roots
Once you have isolated the square root term, the next step involves performing operations to simplify the equation further.

In our problem, after isolating the square root term, we have: 2y^{1/2} = 20.
The goal is to get rid of the coefficient (in this case, the number 2) in front of the square root.

To do this, we divide both sides of the equation by 2: 2y^{1/2} / 2 = 20 / 2, which simplifies to: y^{1/2} = 10.
Now, the square root term is completely isolated, allowing us to proceed to the next step.

Squaring Both Sides
After isolating the square root term and simplifying through operations, the final step is to square both sides of the equation.

Squaring both sides will help to eliminate the square root, making it easier to solve for the variable.

For our example, we have: y^{1/2} = 10.
To find y, we square both sides of the equation: (y^{1/2})^2 = 10^2.
which simplifies to: y = 100.
This final step gives us the value of y, which we found to be 100.

Always remember to check your solution by plugging it back into the original equation. In our case, substituting y = 100 into the original equation confirms that our solution is correct.

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