Problem 46 - Intermediate Algebra

Question

Solve. \(\sqrt{7 x-4}=\sqrt{4-7 x}\)

Step-by-Step Solution

Answer

The solution is \(x = \frac{4}{7}\).

Step 1: Square Both Sides

To eliminate the square roots, square both sides of the equation: \[(\sqrt{7x - 4})^2 = (\sqrt{4 - 7x})^2\]This simplifies to:\[7x - 4 = 4 - 7x\]

Step 2: Move All x Terms to One Side

Add \(7x\) to both sides to get all \(x\) terms on one side of the equation:\[7x + 7x - 4 = 4\]This simplifies to:\[14x - 4 = 4\]

Step 3: Isolate the Variable Term

Add 4 to both sides to isolate the \(x\) term:\[14x - 4 + 4 = 4 + 4\]This simplifies to:\[14x = 8\]

Step 4: Solve for x

Divide both sides by 14 to solve for \(x\):\[x = \frac{8}{14}\]Simplify the fraction:\[x = \frac{4}{7}\]

Step 5: Verify the Solution

Substitute \(x = \frac{4}{7}\) back into the original equation to verify:\[\sqrt{7 \times \frac{4}{7} - 4} = \sqrt{4 - 7 \times \frac{4}{7}}\]\[\sqrt{4 - 4} = \sqrt{4 - 4}\]Both sides equal \(\sqrt{0} = \sqrt{0}\), which is true. Thus, \(x = \frac{4}{7}\) is a valid solution.

Key Concepts

Squaring Both Sides

To solve an equation involving square roots like \(\sqrt{7x - 4} = \sqrt{4 - 7x}\), the first step is to eliminate the square roots. A powerful method to achieve this is squaring both sides of the equation. When you square a square root, you effectively "undo" the square root. For example, squaring \((\sqrt{a})\) gives you \(a\). This means:

• From \( (\sqrt{7x - 4})^2 = (\sqrt{4 - 7x})^2 \), we get \( 7x - 4 = 4 - 7x \).

This process simplifies the equation, removing the square roots, and yields an equation that's much easier to manipulate further. But remember, squaring can sometimes introduce extraneous solutions, so verification is key, which we'll discuss later.

Isolating Variables

Once the square roots are removed, the next goal is to gather all terms involving the variable \(x\) on one side. In our example, we have:

• First, add \(7x\) to both sides: \(7x - 4 + 7x = 4\), which simplifies to \(14x - 4 = 4\).

This step is crucial as it consolidates like terms, allowing us to better isolate the desired variable. It's like organizing a stack of papers where you keep only what you need in one pile. Next comes isolating the terms so we can solve for \(x\) directly, as explained in the next section.

Equation Verification

After you solve for the variable, it's important to verify the solution by substituting it back into the original equation. This ensures that it satisfies both sides of the equation. For the equation \(\sqrt{7x - 4} = \sqrt{4 - 7x}\) with solution \(x = \frac{4}{7}\), we substitute back to check:

• Left side: \(\sqrt{7 \times \frac{4}{7} - 4} = \sqrt{4 - 4} = \sqrt{0}\).
• Right side: \(\sqrt{4 - 7 \times \frac{4}{7}} = \sqrt{4 - 4} = \sqrt{0}\).

Both sides equate to \(\sqrt{0}\), confirming that \(x = \frac{4}{7}\) is indeed a valid solution.

Simplifying Fractions

Sometimes, after solving for the variable, you end up with a fraction that needs to be simplified. Simplifying is the process of reducing the fraction to its lowest terms. For the fraction \(\frac{8}{14}\), we simplify by finding the greatest common divisor (GCD) of 8 and 14:

• The GCD of 8 and 14 is 2.
• Divide both the numerator and the denominator by 2 to get \(\frac{4}{7}\).

This step ensures the solution is presented in its simplest form, making it easier to interpret and use in further applications.