Problem 67 - Algebra A Combined Function

Question

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

Step-by-Step Solution

Answer

The solution is \( x = \frac{11}{5} \).

Step 1: Isolate the Square Root

The original equation is \( \sqrt{5x - 2} = 3 \). Since the square root is already isolated on one side of the equation, we can proceed to the next step.

Step 2: Square Both Sides

To eliminate the square root, square both sides of the equation, which gives \( (\sqrt{5x - 2})^2 = 3^2 \). This simplifies to \( 5x - 2 = 9 \).

Step 3: Solve for x

Now, solve for \( x \). First, add 2 to both sides: \( 5x - 2 + 2 = 9 + 2 \), resulting in \( 5x = 11 \). Then divide both sides by 5: \( x = \frac{11}{5} \).

Step 4: Verify the Solution

Substitute \( x = \frac{11}{5} \) back into the original equation to check your work: \( \sqrt{5(\frac{11}{5}) - 2} = \sqrt{11 - 2} = \sqrt{9} = 3 \). The left side equals the right side, so \( x = \frac{11}{5} \) is the correct solution.

Key Concepts

Square Root Equations

Square root equations involve finding the value of a variable that is under a square root symbol. To solve square root equations, it's often necessary to eliminate the square root by performing algebraic manipulations.
In the given problem, the equation is \( \sqrt{5x - 2} = 3 \). Our goal is to find the value of \( x \) that satisfies this equation. Square root equations can initially look intimidating because of the radical sign, but they can become straightforward once we understand the process of removing the square root.

• Start by ensuring the square root term is isolated on one side, which indicates no other numbers or variables besides the square root and the equals sign.
• Once isolated, the next step is to square both sides of the equation. This removes the square root, allowing us to focus solely on solving the resulting linear equation.
• Remember that squaring both sides of an equation is a legal operation, but it requires careful handling as it can sometimes introduce extraneous solutions.

Isolation of Terms

Isolation of terms is a fundamental strategy in algebra to simplify and solve equations. It involves rearranging the equation so a particular term or variable stands alone on one side of the equation.
In this problem, we dealt with the equation \( 5x - 2 = 9 \) after having squared both sides to remove the square root. The logic behind isolation is to systematically undo operations around the variable:

• First, address any subtraction or addition directly affecting the isolated term. In our equation, adding 2 on both sides gives us \( 5x = 11 \).
• Next, resolve any multiplication or division. Here, dividing both sides by 5 gives us \( x = \frac{11}{5} \).

By going through these steps, we focus entirely on the variable of interest, simplifying the process and helping ensure our solution is correct. This strategy is powerful in achieving clarity and accuracy in solving equations.

Verification of Solutions

Verification is an essential step in solving equations, ensuring the solution is accurate and consistent with the original problem. Especially for square root equations, checking the solution can confirm its validity.
In this exercise, after finding \( x = \frac{11}{5} \), inserting it back into the original square root equation can validate our solution:

• Substitute \( x \) with \( \frac{11}{5} \) to get \( \sqrt{5(\frac{11}{5}) - 2} \).
• By simplifying, we have \( \sqrt{11 - 2} = \sqrt{9} = 3 \), which matches the right side of the equation, confirming it works.

This process of plugging the value back into the original equation is crucial, as it not only confirms the correctness but also reassures that no mistakes were made in earlier steps, such as during squaring or isolating terms. Moreover, it helps identify any extraneous solutions that may have arisen during squaring.