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Chapter 3: Problem 50

Solve. $$\sqrt{3 x-1}+2=7$$

Short Answer

Expert verified
The solution is \( x = \frac{26}{3} \).

Step by step solution

01

Isolate the Square Root

First, isolate the square root term on one side of the equation. Subtract 2 from both sides: \[ \sqrt{3x - 1} + 2 = 7 \] Subtract 2 from both sides: \[ \sqrt{3x - 1} = 5 \]
02

Square Both Sides

To eliminate the square root, square both sides of the equation: \[ (\sqrt{3x - 1})^2 = 5^2 \] This simplifies to: \[ 3x - 1 = 25 \]
03

Solve for x

Now, solve for \( x \). Add 1 to both sides: \[ 3x - 1 + 1 = 25 + 1 \] This gives: \[ 3x = 26 \] Now, divide by 3: \[ x = \frac{26}{3} \]
04

Verify the Solution

Finally, substitute \( x \) back into the original equation to verify: \[ \sqrt{3(\frac{26}{3}) - 1} + 2 = 7 \] Simplify inside the square root: \[ \sqrt{26 - 1} + 2 = 7 \] \[ \sqrt{25} + 2 = 7 \] Since \( 5 + 2 = 7 \), the solution is correct.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Isolating Variables
Isolating variables means getting the variable you want to solve for by itself on one side of the equation. In the given problem, we start by isolating the square root term. The first step is to subtract 2 from both sides of the equation:

\( \sqrt{3x - 1} + 2 = 7 \)
\( \sqrt{3x - 1} = 5 \)
By subtracting 2 from both sides, we simplify our work and can focus on removing the square root next. This is crucial because it prepares us for the next steps where we deal with the square root directly.
Square Both Sides
Once the square root is isolated, the next step is to square both sides of the equation. This will eliminate the square root and simplify the equation further. Squaring both sides involves raising each side of the equation to the power of 2:

\( (\sqrt{3x - 1})^2 = 5^2 \)
This simplifies to: \( 3x - 1 = 25 \).
By squaring the square root, we essentially remove it, making our equation a linear one, easier to solve.
Verify the Solution
This step is all about ensuring our solution is correct. Substitute the found value of \( x \) back into the original equation to check if both sides are equal. Substitute \( x = \frac{26}{3} \) into the original equation:

\( \sqrt{3(\frac{26}{3}) - 1} + 2 = 7 \)
Simplify inside the square root:
\( \sqrt{26 - 1} + 2 = 7 \)
\( \sqrt{25} + 2 = 7 \)
Since \( 5 + 2 = 7 \), our solution is correct. This final check ensures that the solution not only satisfies our transformed equation but also the original problem.

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