Problem 403 - Elementary Algebra

Question

In the following exercises, solve. \(\sqrt{6 n+1}+4=8\)

Step-by-Step Solution

Answer

n = \frac{5}{2}

Step 1: Isolate the Square Root Term

Subtract 4 from both sides of the equation to isolate the square root term:\( \begin{aligned} \sqrt{6n + 1} + 4 - 4 &= 8 - 4 \ \sqrt{6n + 1} &= 4 \ \end{aligned} \)

Step 2: Square Both Sides

Square both sides of the equation to eliminate the square root:\( \begin{aligned} \left(\sqrt{6n + 1}\right)^2 &= 4^2 \ 6n + 1 &= 16 \ \end{aligned} \)

Step 3: Solve for n

Subtract 1 from both sides, then divide by 6 to solve for n:\( \begin{aligned} 6n + 1 - 1 &= 16 - 1 \ 6n &= 15 \ n &= \frac{15}{6} \ n &= \frac{5}{2}\ \ \end{aligned} \)

Key Concepts

Isolating the Square Root

To solve an equation involving a square root like \(\sqrt{6n + 1} + 4 = 8\), our first goal is to isolate the square root term. This means we want to get the square root by itself on one side of the equation. Here's how we do it step by step:

• Start with the original equation: \(\sqrt{6n + 1} + 4 = 8\).
• Subtract 4 from both sides of the equation to cancel the 4 on the left side.
• This gives us: \(\sqrt{6n + 1} = 8 - 4\).
• Simplify the right side: \(\sqrt{6n + 1} = 4\).

By isolating the square root, we make it easier to remove the square root in the next step. This is essential for making the equation simpler to solve.

Squaring Both Sides

Once we have isolated the square root term, the next step is to get rid of the square root. We do this by squaring both sides of the equation. Squaring both sides will remove the square root but be careful to perform this step accurately. Here is how it's done:

• Start from the simplified equation: \(\sqrt{6n + 1} = 4\).
• Square both sides of the equation. On the left side, squaring a square root cancels out the root: \((\sqrt{6n + 1})^2\).
• You'll get: \(6n + 1 = 4^2\).
• On the right side, \(4^2 = 16\).

So, after squaring both sides, we are left with a simpler algebraic equation: \(6n + 1 = 16\). This gets rid of the square root and makes the final steps of solving the equation straightforward.

Algebraic Manipulation

Now that we have a straightforward algebraic equation, we can use simple algebraic manipulation to solve for the variable. Follow these steps:

• Start with the equation: \(6n + 1 = 16\).
• First, subtract 1 from both sides to cancel out the 1 on the left: \(6n = 16 - 1\).
• Simplify it to \(6n = 15\).
• Next, divide both sides by 6 to solve for \(n\): \(n = \frac{15}{6}\).
• Simplify the fraction: \(n = \frac{5}{2}\).

By performing these algebraic manipulations, we find that \(n = \frac{5}{2}\) is the solution to the original equation \(\sqrt{6n + 1} + 4 = 8\).