Problem 112 - Introductory Algebra for College Students

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$x-\frac{1}{5}=\frac{4}{5} x$$

Step-by-Step Solution

Answer

The statement is false. The correct equation should be \(x = \frac{5}{3}\)

Step 1: Analyze and Rearrange

Analyze the statement by isolating \(x\) on one side of the equation. This is done by subtracting \(\frac{1}{5} x\) from both sides of the equation, leading to \(\frac{4}{5} x - \frac{1}{5} x = 1\).

Step 2: Simplify

Simplify the left side of the equation to come up with \(\frac{3}{5} x = 1\).

Step 3: Solve for \(x\)

To further isolate \(x\), divide both sides of the equation by \(\frac{3}{5}\), or multiply them by \(\frac{5}{3}\), to find that \(x = \frac{5}{3}\).

Step 4: Verify and correct

Substitute \(x\) into the original equation : \(x - \frac{1}{5} = \frac{4}{5} x\). Substituting gives \( \frac{5}{3} - \frac{1}{5} = \frac{4}{5} * \frac{5}{3}\). Which simplifies to \(\frac{5}{3} - \frac{1}{5} = \frac{4}{3}\). It results in a statement that is not true. Thus, the original statement \(x - \frac{1}{5} = \frac{4}{5} x\) is false. The correct statement should be \( x = \frac{5}{3}\).