Question

Find all values of x such that y equals 0.
y equals StartFraction x plus 4 Over 7 x minus 35 EndFraction minus StartFraction 3 Over x minus 5 EndFraction minus two sevenths
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Part 1
Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.
A.
The​ value(s) of x for which y equals 0 ​is/are StartSet nothing EndSet .
  
​(Type an integer or a simplified fraction. Use a comma to separate answers as​ needed.)
B.
y equals 0 for all values of x.
C.
There are no values of x for which y equals 0.

Answer 1

To find the values of \( x \) for which \( y = 0 \), we need to solve the equation:

\[
\frac{x + 4}{7x - 35} - \frac{3}{x - 5} - \frac{2}{7} = 0
\]

First, simplify the components of the equation.

1. The term \( \frac{x + 4}{7x - 35} \) can be factored in its denominator as:
\[
7x - 35 = 7(x - 5)
\]
So, the equation becomes:
\[
\frac{x + 4}{7(x - 5)} - \frac{3}{x - 5} - \frac{2}{7} = 0
\]

2. To simplify the equation, we will find a common denominator, which will be \( 7(x - 5) \):
\[
y = \frac{x + 4}{7(x - 5)} - \frac{3 \cdot 7}{7(x - 5)} - \frac{2(x - 5)}{7(x - 5)} = 0
\]

3. Now, rewrite the equation using the common denominator:
\[
\frac{x + 4 - 21 - 2(x - 5)}{7(x - 5)} = 0
\]

4. Combine and simplify the numerator:
\[
x + 4 - 21 - 2x + 10 = -x - 7
\]
Thus, we have:
\[
\frac{-x - 7}{7(x - 5)} = 0
\]

5. For a fraction to equal zero, the numerator must equal zero (the denominator cannot be zero):
\[
-x - 7 = 0
\]
Solving this gives:
\[
-x = 7 \quad \Rightarrow \quad x = -7
\]

Next, we need to ensure that \( x = -7 \) does not make the denominator zero:
- The denominator \( 7(x - 5) \) is zero when \( x = 5 \). Since \( -7 \neq 5 \), this value is valid.

Thus, the only solution for \( y = 0 \) is:

\[
x = -7
\]

**Final Answer:**
The value(s) of \( x \) for which \( y = 0 \) is/are \( \{-7\} \).