Problem 398 - Elementary Algebra

Question

In the following exercises, solve. \(\sqrt{2 n-1}-3=0\)

Step-by-Step Solution

Answer

n = 5

Step 1: Isolate the Square Root

The equation given is \(\sqrt{2n - 1} - 3 = 0\). To isolate the square root term, add \3\ to both sides of the equation: \(\sqrt{2n - 1} = 3\).

Step 2: Square Both Sides

To eliminate the square root, square both sides of the equation: \[ \left(\sqrt{2n - 1}\right)^2 = 3^2 \] This simplifies to: \[2n - 1 = 9\]

Step 3: Solve for n

Now, solve the linear equation for \. First, add 1 to both sides: \[2n - 1 + 1 = 9 + 1\] Simplifying, we get: \[2n = 10\] Then, divide both sides by 2 to find \: \[n = 5\]

Key Concepts

Isolating the Square Root

When you start solving radical equations, the first step is often isolating the square root. This means getting the square root term all by itself on one side of the equation.
Let’s break down our example: \ \(\sqrt{2n - 1} - 3 = 0\)
Here, the square root term \(\sqrt{2n - 1}\) is being subtracted by 3. We need to move -3 to the other side of the equation.
To do this, add 3 to both sides of the equation:
\[ \sqrt{2n - 1} = 3 \]
This step is crucial because having the square root isolated lets you take the next step of eliminating the square root more easily.

Eliminating the Square Root

Eliminating the square root is the next big step. Once you have the square root isolated, it’s time to remove it.
In our example, we isolated the square root to get:
\[ \sqrt{2n - 1} = 3 \]
To eliminate the square root, you need to square both sides of the equation. This is because squaring a square root will leave you with just the expression inside.
Square both sides:
\[ \left(\sqrt{2n - 1}\right)^2 = 3^2 \]
This simplifies to:
\[ 2n - 1 = 9 \]
Now, the equation is free of any square roots and is much simpler to solve.

Solving Linear Equations

After eliminating the square root, you’re often left with a linear equation, which is generally easier to solve.\

In our example, we have:
\[ 2n - 1 = 9 \]
Solving a linear equation usually involves isolating the variable. Start by getting rid of the constant term on the side with the variable. Add 1 to both sides:
\[ 2n - 1 + 1 = 9 + 1 \]
This simplifies to:
\[ 2n = 10 \]
Now, to solve for \(n\), divide both sides by 2:
\[ \frac{2n}{2} = \frac{10}{2} \]
So, we get:
\[ n = 5 \]
And that’s our solution! Every step builds on the previous one, from isolating the square root to eliminating it and finally solving the linear equation. Take your time with each step to make sure you understand what’s happening.