/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 13 Solve each equation. $$ \sqr... [FREE SOLUTION] | 91影视

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Chapter 10: Problem 13

Solve each equation. $$ \sqrt{3 x+1}-4=0 $$

Short Answer

Expert verified
The solution is \( x = 5 \).

Step by step solution

01

Isolate the Square Root

To solve the equation, first isolate the square root term by adding 4 to both sides of the equation: \ \( \sqrt{3x + 1} - 4 = 0 \) \ becomes \ \( \sqrt{3x + 1} = 4 \)
02

Square both Sides

Next, eliminate the square root by squaring both sides of the equation: \ \( (\sqrt{3x + 1})^2 = 4^2 \) \ which simplifies to \ \( 3x + 1 = 16 \)
03

Solve for x

Now, solve for x by isolating it. First, subtract 1 from both sides of the equation: \ \( 3x + 1 - 1 = 16 - 1 \) \ which simplifies to \ \( 3x = 15 \) Then, divide both sides by 3: \ \( x = 5 \)

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Key Concepts

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

Isolating the Square Root
To solve radical equations, the first thing you need to do is isolate the square root term. This means getting the square root expression \(\sqrt{3x + 1}\) on one side of the equation by itself. In the given equation \(\sqrt{3x + 1} - 4 = 0\), we can isolate the square root by adding 4 to both sides. This operation effectively moves the -4 to the right side of the equation. Therefore:
\[ \sqrt{3x + 1} = 4 \]
This step is crucial because it sets up the equation for the next move, which is to eliminate the square root entirely.
Squaring Both Sides
Once the square root is isolated, the next step is to square both sides of the equation. Squaring undoes the square root, transforming \(\sqrt{3x + 1} = 4\) into a simpler form. Here鈥檚 what it looks like:
\[ (\sqrt{3x + 1})^2 = 4^2 \]
This simplifies to:
\[ 3x + 1 = 16 \]
By squaring both sides, you鈥檝e converted the equation into a linear format that is much easier to handle. Keep in mind, though, squaring can sometimes introduce extraneous solutions, so always check your final results.
Solving for x
Now that we have a linear equation \(3x + 1 = 16\), the task is to solve for x. First, subtract 1 from both sides to move the constant term:
\[ 3x + 1 - 1 = 16 - 1 \]
This simplifies to:
\[ 3x = 15 \]
Now, divide both sides of the equation by 3 to isolate x:
\[ x = 5 \]
By following these steps, you have found that the solution to the original radical equation is \(x = 5\). Always remember to verify your solution by plugging it back into the original equation to ensure it holds true.

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