Problem 12 - Intermediate Algebra

Question

For Problems 1-56, solve each equation. Don't forget to check each of your potential solutions. $$ \sqrt{3 y+5}-2=0 $$

Step-by-Step Solution

Answer

The solution is \( y = -\frac{1}{3} \).

Step 1: Isolate the square root

To start solving the equation, we need to isolate the square root term. Our equation is: \( \sqrt{3y + 5} - 2 = 0 \). Add 2 to both sides to get: \( \sqrt{3y + 5} = 2 \).

Step 2: Square both sides

Remove the square root by squaring both sides of the equation: \( (\sqrt{3y + 5})^2 = 2^2 \). This simplifies to: \( 3y + 5 = 4 \).

Step 3: Solve for y

To isolate \( y \), subtract 5 from both sides: \( 3y + 5 - 5 = 4 - 5 \), resulting in \( 3y = -1 \). Now, divide both sides by 3: \( y = -\frac{1}{3} \).

Step 4: Check the solution

Substitute \( y = -\frac{1}{3} \) back into the original equation to verify. The original equation is \( \sqrt{3y + 5} - 2 = 0 \). Substituting gives \( \sqrt{3(-\frac{1}{3}) + 5} - 2 = \sqrt{-1 + 5} - 2 = \sqrt{4} - 2 = 2 - 2 = 0 \), which is true.

Key Concepts

Square Root Equations

Square root equations involve solving for a variable that is wrapped inside a square root symbol \( \sqrt{} \). These equations might seem complex at first, but they follow a logical sequence of steps. The goal is to isolate the square root expression on one side of the equation by itself.
This is done by performing algebraic operations such as adding, subtracting, multiplying, or dividing. Once isolated, you can eliminate the square root by squaring both sides of the equation. Thus, square root equations often transform into simpler linear or quadratic forms that are more straightforward to solve.

Equation Solving Steps

Solving square root equations methodically involves several key steps. Firstly, you should isolate the square root term. For example, with the equation \( \sqrt{3y + 5} - 2 = 0 \), you would add 2 to both sides. This isolates the square root, resulting in \( \sqrt{3y + 5} = 2 \).
Next, eliminate the square root by squaring both sides of the equation. By doing so, you get \( 3y + 5 = 4 \). This simplifies the equation to a linear form that can be solved using basic algebraic manipulation.
Finally, solve for \( y \) by isolating it on one side. Subtract 5 from both sides to get \( 3y = -1 \), and then divide by 3, resulting in \( y = -\frac{1}{3} \).

• Step 1: Isolate the square root.
• Step 2: Square both sides to remove the square root.
• Step 3: Solve the resulting linear equation for the variable.

Checking Solutions

Checking your solutions is a crucial step in verifying the correctness of your answer, especially when dealing with square root equations. Sometimes, squaring both sides of an equation can introduce extraneous solutions—answers that don't actually satisfy the original equation.
To verify that your solution is correct, substitute it back into the original equation and check if the left-hand side equals the right-hand side. For the equation \( \sqrt{3y + 5} - 2 = 0 \), test \( y = -\frac{1}{3} \) by substituting it back in: \( \sqrt{3(-\frac{1}{3}) + 5} - 2 = \sqrt{4} - 2 = 0 \), which holds true.

• Substitute the solution back to see if the equation balances.
• Confirm that no extra solutions were inadvertently introduced.

By doing these checks, you ensure your solution is valid and accurate.