Problem 16 - Algebra 2

Question

Solve each equation. $$ 5+\sqrt{4 y-5}=12 $$

Step-by-Step Solution

Answer

The solution is \(y = 13.5\).

Step 1: Isolate the square root term

To start solving the equation, isolate the square root term on one side of the equation. Subtract 5 from both sides to get:\[\sqrt{4y - 5} = 12 - 5\]This simplifies to:\[\sqrt{4y - 5} = 7\]

Step 2: Eliminate the square root

To remove the square root, square both sides of the equation. This gives us:\[(\sqrt{4y - 5})^2 = 7^2\]which simplifies to:\[4y - 5 = 49\]

Step 3: Solve for the variable

Now, solve for \(y\) by first adding 5 to both sides:\[4y - 5 + 5 = 49 + 5\]This results in:\[4y = 54\]Next, divide both sides by 4:\[y = \frac{54}{4}\]which simplifies to:\[y = 13.5\]

Step 4: Verify the solution

Substitute \(y = 13.5\) back into the original equation to ensure it satisfies the equation:\[5 + \sqrt{4(13.5) - 5} = 12\]Calculate inside the square root:\[4(13.5) - 5 = 54 - 5 = 49\]which gives:\[5 + \sqrt{49} = 5 + 7 = 12\]Since the original equation holds true, \(y = 13.5\) is a valid solution.

Key Concepts

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying equations to make them easier to solve. In the given exercise, our goal is to isolate the square root term. This is a common step in algebra when dealing with equations that involve radicals or roots.

• Firstly, identify which part of the equation is the square root term, and gauge how to isolate it. In our exercise, the term is \( \sqrt{4y - 5} \).
• We start by removing any constants on the same side as the square root. For example, by subtracting 5, we simplify the equation to \( \sqrt{4y - 5} = 7 \).
• This reorganization makes it simpler to apply further mathematical operations, like eliminating the roots in subsequent steps.

The initial manipulation sets the stage for isolating variables, making it easier to solve for them later.

Square Roots

Understanding square roots is crucial when working with equations like the one in our exercise. A square root asks the question: "What number multiplied by itself gives this number?"

• To eliminate a square root, square both sides of the equation. This means if \( \sqrt{a} = b \), then \( a = b^2 \).
• In the exercise, squaring \( \sqrt{4y - 5} = 7 \) yields \( 4y - 5 = 49 \). This removes the radical, allowing us to solve for \( y \) more straightforwardly.
• It's important to remember that when squaring, you change the function's properties, so always verify solutions afterward.

Thus, squaring helps eliminate the roots, but it's essential to handle these operations carefully to avoid errors.

Verifying Solutions

Verifying solutions is an essential step to ensure that the solution derived actually satisfies the original equation. In solving equations, operations like squaring might introduce extraneous solutions, so verification is crucial.

• The verification process involves substituting the obtained value back into the original equation. For our variable \( y = 13.5 \), replace it in: \( 5 + \sqrt{4(13.5) - 5} = 12 \).
• Calculate the expression within the square root: \( 4 \times 13.5 - 5 = 49 \). Compute the square root of 49, which is 7.
• Verify if the left side equals the right side of the equation: \( 5 + 7 = 12 \). Since both sides are equal, the solution is correct.

Verification helps identify any potential errors during calculation and reassures that the solution truly solves the initial problem.