Question

Passing through left parenthesis negative 1 comma 9 right parenthesis and parallel to the line whose equation is 5 x minus 6 y minus 7 equals 0

find equation

Answer 1

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

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

2. **Since parallel lines have the same slope**, the slope of the line we want to find is also \(\frac{5}{6}\).

3. **Use the point-slope form of the equation of a line**, which is given by:
\[
y - y_1 = m(x - x_1)
\]
Here, \((x_1, y_1)\) is the point \((-1, 9)\) and \(m = \frac{5}{6}\). Substituting these values in:
\[
y - 9 = \frac{5}{6}(x + 1)
\]

4. **Simplifying this equation**:
\[
y - 9 = \frac{5}{6}x + \frac{5}{6}
\]
Adding 9 to both sides:
\[
y = \frac{5}{6}x + \frac{5}{6} + 9
\]

To combine \(\frac{5}{6}\) and \(9\), we can express 9 as \(\frac{54}{6}\):
\[
y = \frac{5}{6}x + \frac{5 + 54}{6} = \frac{5}{6}x + \frac{59}{6}
\]

5. **Convert to standard form (Ax + By + C = 0)**:
Start with:
\[
y = \frac{5}{6}x + \frac{59}{6}
\]
Multiply through by 6 to eliminate the fraction:
\[
6y = 5x + 59
\]
Rearranging gives:
\[
-5x + 6y - 59 = 0
\]
Or:
\[
5x - 6y + 59 = 0
\]

Thus, the equation of the line passing through the point \((-1, 9)\) and parallel to the line \(5x - 6y - 7 = 0\) is \(\boxed{5x - 6y + 59 = 0}\).