Question

Passing through left parenthesis negative 1 comma 7 right parenthesis and parallel to the line whose equation is 7 x minus 6 y minus 5 equals 0
The equation of the line is
  
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Answer 1

To find the equation of a line that passes through the point \((-1, 7)\) and is parallel to the line given by the equation \(7x - 6y - 5 = 0\), we first need to determine the slope of the given line.

1. **Rewrite the equation in slope-intercept form (y = mx + b):**
\[
7x - 6y - 5 = 0
\]
Rearranging gives:
\[
-6y = -7x + 5
\]
\[
y = \frac{7}{6}x + \frac{5}{6}
\]
From this, we can see that the slope \(m\) of the line is \(\frac{7}{6}\).

2. **Use the point-slope form of the line:**

The point-slope form of the equation of a line is:
\[
y - y_1 = m(x - x_1)
\]
where \((x_1, y_1)\) is the point the line passes through, and \(m\) is the slope.

Substituting the point \((-1, 7)\) for \((x_1, y_1)\) and \(m = \frac{7}{6}\):
\[
y - 7 = \frac{7}{6}(x + 1)
\]

3. **Simplify the equation:**
Distributing \(\frac{7}{6}\):
\[
y - 7 = \frac{7}{6}x + \frac{7}{6}
\]
Adding \(7\) (or \(\frac{42}{6}\)) to both sides:
\[
y = \frac{7}{6}x + \frac{7}{6} + \frac{42}{6}
\]
Combining the terms on the right:
\[
y = \frac{7}{6}x + \frac{49}{6}
\]

4. **Convert back to standard form (optional):**
Multiplying through by \(6\) to eliminate the fraction:
\[
6y = 7x + 49
\]
Rearranging:
\[
-7x + 6y - 49 = 0
\]
or
\[
7x - 6y + 49 = 0
\]

Thus, the equation of the line that passes through the point \((-1, 7)\) and is parallel to the line \(7x - 6y - 5 = 0\) is:
\[
\boxed{7x - 6y + 49 = 0}
\]