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.
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The term \( \frac{x + 4}{7x - 35} \) can be factored in its denominator as: \[ 7x - 35 = 7(x - 5) \] So, the equation becomes: \[ \frac{x + 4}{7(x - 5)} - \frac{3}{x - 5} - \frac{2}{7} = 0 \]
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To simplify the equation, we will find a common denominator, which will be \( 7(x - 5) \): \[ y = \frac{x + 4}{7(x - 5)} - \frac{3 \cdot 7}{7(x - 5)} - \frac{2(x - 5)}{7(x - 5)} = 0 \]
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Now, rewrite the equation using the common denominator: \[ \frac{x + 4 - 21 - 2(x - 5)}{7(x - 5)} = 0 \]
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Combine and simplify the numerator: \[ x + 4 - 21 - 2x + 10 = -x - 7 \] Thus, we have: \[ \frac{-x - 7}{7(x - 5)} = 0 \]
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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 \( {-7} \).