Problem 38 - Algebra A Combined Function

Question

Solve. $$ \sqrt{3 y+6}=\sqrt{7 y-6} $$

Step-by-Step Solution

Answer

The solution is \( y = 3 \).

Step 1: Understand the Equation

We are given the equation \( \sqrt{3y + 6} = \sqrt{7y - 6} \). Our goal is to solve for \( y \). The equation involves square roots, so we will need to eliminate them to solve for \( y \).

Step 2: Eliminate the Square Roots

To remove the square roots, square both sides of the equation: \[(\sqrt{3y + 6})^2 = (\sqrt{7y - 6})^2\]which simplifies to:\[3y + 6 = 7y - 6\]

Step 3: Rearrange the Terms

Rearrange the equation to get all terms involving \( y \) on one side and constant terms on the other side. Subtract \( 3y \) from both sides:\[6 = 4y - 6\]Then add 6 to both sides:\[12 = 4y\]

Step 4: Solve for y

To find the value of \( y \), divide both sides by 4:\[y = \frac{12}{4}\]This simplifies to:\[y = 3\]

Step 5: Verify the Solution

Substitute \( y = 3 \) back into the original equation to check:\[\sqrt{3(3) + 6} = \sqrt{7(3) - 6}\]which simplifies to:\[\sqrt{9 + 6} = \sqrt{21 - 6}\]\[\sqrt{15} = \sqrt{15}\]Since both sides are equal, the solution \( y = 3 \) is correct.

Key Concepts

Eliminating Square Roots

When solving equations that include square roots, it's often necessary to eliminate these roots to simplify the equation. This is done by squaring both sides, as squaring is the inverse operation of taking a square root. For instance, in the equation \(\sqrt{3y + 6} = \sqrt{7y - 6}\), we square both sides to eliminate the square roots. This process results in the equation \(3y + 6 = 7y - 6\). By handling the equation in this way, we can convert it into a linear form, which is typically easier to solve.

It's important to note that squaring both sides works because if \(a = b\), then \(a^2 = b^2\). During this step, always ensure you apply squaring correctly to avoid errors.

• Square both sides of the equation.
• Check for extraneous solutions that may appear after squaring.

Equation Simplification

Once the square roots are eliminated, the next step is equation simplification. Starting with the equation \(3y + 6 = 7y - 6\), we rearrange the terms to isolate \(y\) on one side of the equation. We do this because we want to solve directly for \(y\). Begin by moving all terms involving \(y\) to one side and the constant terms to the other. This often involves steps such as addition, subtraction, or other basic operations to combine like terms.

In our example, subtracting \(3y\) from both sides and adding 6 leads to the simplified form \(12 = 4y\). Finally, divide by 4 to isolate \(y\), hence resolving to \(y = 3\). Simplifying appropriately is crucial for achieving the correct solution.

• Rearrange terms logically to solve for the variable.
• Perform arithmetic operations carefully to maintain balance.

Verification of Solutions

Verifying your solution is an essential step that ensures correctness. After finding \(y = 3\), substitute this back into the original equation to check if it holds true. In this case, substitute \(y = 3\) into \(\sqrt{3y + 6} = \sqrt{7y - 6}\) and simplify both sides to confirm they equal.

When plugged back in, both sides of the equation simplify to \(\sqrt{15}\), confirming that our solution is accurate. This verification step is valuable because it also helps identify any extraneous solutions that might have been introduced during squaring.

• Always verify by substituting the solution back into the original equation.
• Ensure both sides of the equation are equal after substitution.

If they match, your solution is correct; if not, reevaluate your steps.