Problem 23 - Algebra A Combined Function

Question

Solve. $$ \sqrt{3 x-2}=5 $$

Step-by-Step Solution

Answer

x = 9

Step 1: Understand the Equation

The given equation is \( \sqrt{3x - 2} = 5 \). This is a square root equation where the expression under the square root needs to be isolated.

Step 2: Eliminate the Square Root

To eliminate the square root, square both sides of the equation: \((\sqrt{3x - 2})^2 = 5^2\). This simplifies to \(3x - 2 = 25\).

Step 3: Solve for 3x

Rearrange the equation to solve for \(3x\). Add 2 to both sides: \(3x = 25 + 2\). This simplifies to \(3x = 27\).

Step 4: Solve for x

Divide both sides by 3 to isolate \(x\): \(x = \frac{27}{3}\). This simplifies to \(x = 9\).

Step 5: Check Your Solution

Verify the solution by substituting \(x = 9\) back into the original equation: \(\sqrt{3(9) - 2} = \sqrt{27 - 2} = \sqrt{25} = 5\). The left-hand side equals the right-hand side, confirming that the solution is correct.

Key Concepts

Square Root Equations

Square root equations are a type of equation where the variable is under a square root. These equations can initially appear complex, but they become manageable with the right approach. The key to solving square root equations is isolating the square root expression on one side of the equation.
Once isolated, you can eliminate the square root by squaring both sides of the equation.

• This process involves raising both the square root and the other side of the equation to the power of two.
• Squaring both sides cancels out the square root, leaving the expression inside free from the root.

This crucial step turns a challenging square root equation into a simpler equation you can solve using basic algebraic techniques. But remember, always check for extraneous solutions that may arise from squaring both sides.

Equation Solving Techniques

When it comes to solving square root equations, transforming them into linear equations is a powerful strategy. After isolating and eliminating the square root, what's left is usually a linear equation. Solving these is straightforward:

• Rearrange the equation to isolate terms containing the variable on one side.
• Combine like terms and simplify the equation as needed.
• Solve for the variable by performing inverse operations—addition, subtraction, multiplication, or division.

In our example, after isolating and eliminating the square root, we had the equation 3x - 2 = 25. Adding 2 to both sides gave 3x = 27, and dividing both sides by 3 isolated x. This step-by-step technique makes equation solving systematic and reliable.

Verification of Solutions

Verifying solutions to equations is an essential step in the problem-solving process. Even when all operations seem correct, we must ensure that our solution satisfies the original equation. Here's how you can verify solutions:

• Substitute the computed value back into the original equation.
• Simplify each side of the equation separately to check if both sides are equal.

For our problem, substituting x = 9 back into the equation \( \sqrt{3(9) - 2} = 5 \) shows that both sides equal 5. This confirms that 9 is indeed a valid solution. Verification helps identify errors or extraneous solutions that might arise during the process, ensuring a correct and accurate answer.