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Chapter 0: Problem 115

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 statement is true as the simplified equation is \(0 = 0\).

Step by step solution

01

Break down the equation

Break down the problem into individual fractions, which are:\[\frac{2x - 1}{x - 7}, \quad \frac{3x - 1}{x - 7}, \quad \frac{5x - 2}{x - 7}\]
02

Addition of Fractions

In the problem, fractions are being added and subtracted. Because the denominators of the fractions are the same (\(x - 7\)), you can now add and subtract the numerators: \[\frac{(2x - 1) + (3x - 1) - (5x - 2)}{x - 7}\]
03

Simplify the Fraction

Combine like terms in the numerator to simplify the equation: \[\frac{2x + 3x - 5x - 1 - 1 + 2}{x - 7} = \frac{0}{x - 7}\]
04

Solve the simplified equation

The numerator of the fraction is zero and any number multiplied by zero is zero, so the equation on the left side becomes \(0\). The right side of the equation is also zero. Hence the equation is true.

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Most popular questions from this chapter

Determine whether each statement makes sense or does not make sense, and explain your reasoning. When performing the division $$\frac{7 x}{x+3}+\frac{(x+3)^{2}}{x-5}$$ I began by dividing the numerator and the denominator by the common factor, \(x+3\).

This will help you prepare for the material covered in the next section. Multiply and simplify: \(12\left(\frac{x+2}{4}-\frac{x-1}{3}\right)\).

Explain why \(|x|<-4\) has no solution.

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The equation \((2 x-3)^{2}=25\) is equivalent to \(2 x-3=5\).

$$\text { Solve for } C: \quad V=C-\frac{C-S}{L} N$$.
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