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Chapter 0: Problem 104

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

Short Answer

Expert verified
The given statement is True.

Step by step solution

01

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 \(\frac{a+b}{c}\).
02

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 \(\frac{0}{x-7} = 0\).
03

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.

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