Problem 50 - Beginning and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The solution to the given equation \(\sqrt{3x-1}+2=7\) is \(x = \frac{26}{3}\).

Step 1: Isolate the square root

First, we will isolate the square root by subtracting 2 from both sides of the equation: \(\sqrt{3x - 1} + 2 - 2 = 7 - 2\) I.e., \(\sqrt{3x - 1} = 5\)

Step 2: Square both sides of the equation

Now, we will square both sides of the equation to eliminate the square root. Be careful not to confuse this step with distributing the square inside the square root (which is not allowed): \((\sqrt{3x - 1})^2 = (5)^2\) This simplifies to: \(3x - 1 = 25\)

Step 3: Solve the quadratic equation for x

Now, we will solve the resulting equation: \(3x - 1 - 25 = 0\) I.e., \(3x - 26 = 0\) Now, we will add 26 to both sides to solve for x: \(3x = 26\) Finally, we will divide both sides by 3 to find the value of x: \(x = \frac{26}{3}\) Thus, the solution to the given equation is x = \(\frac{26}{3}\).

Key Concepts

Isolate the Square Root

When solving equations with square roots, the first crucial step is to isolate the square root expression. Think of it as unwrapping a present. You want to get to the heart of the expression where the square root sits. In our exercise, we have

• Your original equation: \(\sqrt{3x - 1} + 2 = 7\)
• First, you need to remove any constants or coefficients outside the square root by using basic operations like addition or subtraction.
• Here, we subtract 2 from both sides: \(\sqrt{3x - 1} = 5\).

Now, the square root is isolated, and you're ready for the next step. Remember, isolating helps simplify the equation, making it easier to solve later.

Squaring Both Sides

Once the square root is isolated, the next step is to eliminate it. This is done by squaring both sides of the equation. Squaring is akin to balancing a scale—what you do to one side, you must do to the other.

• On our exercise: Once you have \(\sqrt{3x - 1} = 5\), you square both sides.
• Be cautious: When you square the square root, it cancels out, leaving the expression inside the root. So \((\sqrt{3x - 1})^2 = 3x - 1\).
• For the right side, \(5^2\) equals 25.
• Thus, you simplify to: \(3x - 1 = 25\).

Squaring effectively removes the square root and transforms your equation into a simpler form, making it much easier to solve.

Solving Linear Equations

Having turned the equation into a linear form, the task now is to solve it. Linear equations are akin to connecting dots; straightforward once the path is clear.

• Your equation \(3x - 1 = 25\) is ready to be solved.
• First, address the constant on the left: Add 1 to both sides to keep balance, giving \(3x = 26\).
• Next, handle the coefficient attached to \(x\) by dividing both sides by 3, resulting in \(x = \frac{26}{3}\).

And there you have your solution! Check your work by substituting \(x = \frac{26}{3}\) back into the original equation to ensure accuracy. Solving linear equations like these often requires just a couple steps and ensures logical progression from problem to solution.