Problem 11 - Algebra for College Students

Question

Solve each equation. Don't forget to check each of your potential solutions. \(\sqrt{4 y-3}-6=0\)

Step-by-Step Solution

Answer

The solution is \(y = 9.75\).

Step 1: Isolate the Square Root

The first step is to isolate the square root term in the equation. We begin with \(\sqrt{4y - 3} - 6 = 0\). Add 6 to both sides to isolate the square root term, resulting in \(\sqrt{4y - 3} = 6\).

Step 2: Eliminate the Square Root

To eliminate the square root, square both sides of the equation. This gives us \((\sqrt{4y - 3})^2 = 6^2\), which simplifies to \(4y - 3 = 36\).

Step 3: Solve for y

Now solve for \(y\) by adding 3 to both sides to get \(4y = 39\). Next, divide both sides by 4 to obtain \(y = \frac{39}{4}\), which simplifies to \(y = 9.75\).

Step 4: Verify the Solution

Finally, check the solution by substituting \(y = 9.75\) back into the original equation to ensure both sides are equal. Substitute to get \(\sqrt{4(9.75) - 3} - 6 = 0\). Simplify inside the square root to \(\sqrt{39 - 3} = \sqrt{36}\). This equals 6, so subtracting 6 gives 0. Hence, \(y = 9.75\) is a valid solution.

Key Concepts

Checking Solutions

After solving the equation, it is crucial to verify that the potential solution satisfies the original equation. Sometimes, resolving an equation can yield extraneous solutions that arise from squaring both sides or other operations.
Substituting the solution back into the original equation helps identify if it works. Here, our original equation is \( \sqrt{4y - 3} - 6 = 0 \). After solving, we propose that \( y = 9.75 \) is a possible solution.
By substituting \( y = 9.75 \) back into the equation, we ensure both sides balance. Plug in and simplify: \( \sqrt{4(9.75) - 3} = 6 \) simplifies to \( \sqrt{36} = 6 \). This confirms that our solution is indeed correct as both sides equal to 6, making it a valid solution.

Isolating Terms

When faced with square root equations, the first significant step often involves isolating the term with the square root to simplify the equation's structure. Here, with \( \sqrt{4y - 3} - 6 = 0 \), the aim is to remove any extraneous constants from the side of the equation with the root.
By adding 6 to both sides, the equation reduces to \( \sqrt{4y - 3} = 6 \), leaving the equation with only the square root term on one side. This simplification is necessary to use other algebraic operations more freely and effectively in the next steps of solving the equation.

Eliminating Square Roots

Once the square root term is isolated, the next goal is to eliminate the square root altogether. This is achieved by squaring both sides of the equation.
This move neutralizes the square root, turning \( \sqrt{4y - 3} = 6 \) into \((\sqrt{4y - 3})^2 = 6^2\). As a result, the equation becomes \( 4y - 3 = 36 \). Squaring ensures we remove the square root, allowing the problem to transform into a simpler linear equation to solve. It's essential to handle this step accurately to avoid errors in the solution process.

Algebraic Manipulation

Algebraic manipulation is the methodical re-arrangement of equations to isolate the variable of interest. With \( 4y - 3 = 36 \), this involves moving terms around to solve for \( y \).
First, add 3 to both sides, resulting in \( 4y = 39 \). Then, divide each side by 4 so that \( y = \frac{39}{4} \), simplifying to \( y = 9.75 \). Each step aims to isolate \( y \) by using basic arithmetic operations, crucial for finding accurate solutions.

• Addition/Subtraction: Used to move constants across the equation.
• Multiplication/Division: Applied to isolate \( y \) completely.

Following these steps closely prevents algebraic errors and ensures the solution is correct.