Problem 104 - College Algebra Essentials

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$\frac{2 x-1}{x-7}+\frac{3 x-1}{x-7}-\frac{5 x-2}{x-7}=0$$

Step-by-Step Solution

Answer

The given statement is True.

Step 1: Remember the properties of fractions

Recall that for fractions with the same denominator, they can be directly added or subtracted. In fact, for \(\frac{a}{c} + \frac{b}{c}\), it equals \(\frac{a+b}{c}\).

Step 2: Apply the property

For the given equation, we see that all three fractions have the same denominator, which is \(x-7\). Hence, we can add the numerators directly using the property mentioned in step 1. So the equation becomes, \(\frac{2x-1+3x-1-5x+2}{x-7} = 0\). Simplify this to get \(\frac{0}{x-7} = 0\).

Step 3: Observe the equation

In the equation \(\frac{0}{x-7} = 0\), we can see that any number times 0 gives 0. Hence, this equation is always true, given that \(x \neq 7\) to avoid division by zero. So, the statement \(\frac{2 x-1}{x-7}+\frac{3 x-1}{x-7}-\frac{5 x-2}{x-7}=0\) is true.