Problem 24 - Intermediate Algebra

Question

Solve. \(\sqrt{5 x-4}=9\)

Step-by-Step Solution

Answer

The solution is \(x = 17\).

Step 1: Rearrange Equation for Squares

The goal is to isolate the variable within its square root. We start by squaring both sides of the equation to eliminate the square root. Thus, we have \((\sqrt{5x-4})^2 = 9^2\).

Step 2: Simplify Both Sides

By simplifying \((\sqrt{5x-4})^2\), we get \(5x - 4\) on the left side. Similarly, \(9^2 = 81\). So the equation becomes \(5x - 4 = 81\).

Step 3: Solve for x

Add 4 to both sides to isolate the term containing \(x\) on one side: \(5x - 4 + 4 = 81 + 4\), which simplifies to \(5x = 85\). Now, divide both sides by 5 to solve for \(x\): \(x = \frac{85}{5} = 17\).

Step 4: Verify the Solution

Substitute \(x = 17\) back into the original equation to verify the solution: \(\sqrt{5 \times 17 - 4}\). Simplifying inside the square root gives \(\sqrt{85 - 4} = \sqrt{81} = 9\), which is correct according to the original equation.

Key Concepts

Isolating Variables

When faced with an equation involving a square root, such as \(\sqrt{5x-4} = 9\), the first step is isolating the variable. Isolation means that we aim to simplify the equation in a way that the variable \(x\) can be easily found.
To do so, we must first remove any operations affecting \(x\) within the square root. Here, it's crucial to move any other terms or operations to the opposite side of the equation.
In our problem, the square root is the main obstacle, so we focus on removing it, which we will address in the next section.

Squaring Both Sides

Squaring both sides of an equation is a powerful technique for eliminating square roots. If we have \(\sqrt{a} = b\), squaring both gives \(a = b^2\).
In our example, squaring \(\sqrt{5x-4} = 9\) results in \((\sqrt{5x-4})^2 = 9^2\). After squaring, the square root on the left hand side disappears, turning into \(5x-4\). Similarly, \(9^2\) simplifies to 81.
So, our equation becomes \(5x - 4 = 81\).
Always remember: when you square both sides, your equation changes dramatically, often making the variable easier to isolate.

Verifying Solutions

Once a solution is found, verifying it ensures accuracy. Always substitute the value back into the original equation to check.
For our solution \(x = 17\), we substitute back into \(\sqrt{5x-4} = 9\).
This involves calculating \(5 \times 17 - 4\), which simplifies to \(85 - 4 = 81\). The square root of 81 is 9, matching the original equation.

• Verification is crucial because squaring both sides might introduce extraneous solutions not valid in the original equation.
• Always perform this step to confirm that your solution is correct, ensuring there's no mistake or assumption mistaken along the way.