Problem 390 - Elementary Algebra

Question

In the following exercises, check whether the given values are solutions. For the equation \(\sqrt{-y+20}=y:\) (a) Is \(y=4\) a solution? (b) Is \(y=-5\) a solution?

Step-by-Step Solution

Answer

Yes, \( y=4 \) is a solution. No, \( y=-5 \) is not a solution.

Step 1: Substitute y = 4 into the equation

Substitute the value of y into the equation \ \ \ \ \( \sqrt{-y+20} = y \) : \( \sqrt{-4 + 20} = 4 \) \. Simplify inside the square root: \( \sqrt{16} = 4 \)

Step 2: Verify the solution for y = 4

Since \( \sqrt{16} = 4 \) is a true statement, \y=4 \ is a solution.

Step 3: Substitute y = -5 into the equation

Substitute the value of y into the equation \( \sqrt{-y + 20} = y \) : \( \sqrt{--5 + 20} = -5 \) \. Simplify inside the square root: \( \sqrt{25} = -5 \)

Step 4: Verify the solution for y = -5

Since \( \sqrt{25} = 5 \) (not -5), \y = -5 \ is not a solution.

Key Concepts

square root equations

Square root equations involve the square root symbol, which looks like this: \(\backslash\backslashsqrt{}\). Solving these equations often requires us to isolate the square root on one side of the equation and then square both sides to eliminate the square root.

For example, let’s consider the equation \(\backslash\backslashsqrt{-y + 20} = y\).

To solve it, we want to first check if potential solutions like \(y=4\) and \(y=-5\) work. We substitute these values into the equation and simplify.

When \(y = 4\):

\(\backslash\backslashsqrt{-4 + 20} = 4\)

This simplifies to \(\backslash\backslashsqrt{16} = 4\), which is true because \(\backslash\backslashsqrt{16} = 4\).

When \(y = -5\):

\(\backslash\backslashsqrt{-(-5) + 20} = -5\)

This simplifies to \(\backslash\backslashsqrt{ 25 } = -5\), but \(\backslash\backslashsqrt{25} = 5\), not \(-5\).

Hence, \(y = 4\) is a solution, but \( y = -5\) is not.

solution verification

Solution verification helps us confirm if our obtained answers are correct. This process ensures you avoid any miscalculations or misinterpretations. After substituting values into the original equation, we simplify the equation and check if both sides are equal.

For the example equation \(\backslash\backslashsqrt{-y+20}=y\):

Substituting \(y = 4\), we got \(\backslash\backslashsqrt{16}=4\), which is a true statement.

Substituting \(y = -5\), we had \(\backslash\backslashsqrt{25}=-5\), which is false because \(\backslash\backslashsqrt{25} = 5\).

This verification step proves that \(y = 4\) is a correct solution, but \(y = -5\) is not.

substitution method

The substitution method involves replacing variables with given values in an equation to check if the equation holds true. This process is crucial for verifying potential solutions.

Using the substitution method for the equation \(\backslash\backslashsqrt{-y+20}=y\):

1. Substitute \(y = 4\):
\(\backslash\backslashsqrt{-4+20} = 4\)
This simplifies to \(\backslash\backslashsqrt{16} = 4\). Since this is true, \(y = 4\) is a solution.

2. Substitute \(y=-5\):
\(\backslash\backslashsqrt{-(-5)+20}=-5\)
This simplifies to \(\backslash\backslashsqrt{25} = -5\), but \(\backslash\backslashsqrt{25} = 5\). Since this is false, \(y = -5\) is not a solution.

The substitution method is a valuable tool for solving and verifying equations. It allows us to plug in potential solutions and confirm their validity easily.