Question

Write an equation in​ slope-intercept form of the line satisfying the given conditions.
Through ​(negative 9​,8​), perpendicular to 4 x minus 9 y equals negative 63

Answer 1

To find the equation of a line in slope-intercept form that is perpendicular to a given line and passes through a specific point, we can follow these steps:

1. **Find the slope of the given line.**
The equation of the given line is:
\[
4x - 9y = -63
\]

First, we can rearrange this equation into slope-intercept form (y = mx + b), where m is the slope.

Start by isolating \( y \):
\[
-9y = -4x - 63
\]
\[
y = \frac{4}{9}x + 7
\]

From this, we see that the slope (m) of the given line is \( \frac{4}{9} \).

2. **Find the slope of the line that is perpendicular to the given line.**
The slope of a line that is perpendicular to another is the negative reciprocal of the original slope. So, the slope of our required line will be:
\[
m_{\text{perpendicular}} = -\frac{1}{(\frac{4}{9})} = -\frac{9}{4}
\]

3. **Use the point-slope form to write the equation of the line.**
We are given the point \((-9, 8)\) through which this line passes. We can use the point-slope form of the equation:
\[
y - y_1 = m(x - x_1)
\]
Substituting the point \((-9, 8)\) and slope \(-\frac{9}{4}\):
\[
y - 8 = -\frac{9}{4}(x + 9)
\]

4. **Distribute and solve for y.**
\[
y - 8 = -\frac{9}{4}x - \frac{9}{4} \times 9
\]
\[
y - 8 = -\frac{9}{4}x - \frac{81}{4}
\]
Now, add 8 to both sides:
\[
y = -\frac{9}{4}x - \frac{81}{4} + 8
\]
Converting 8 to a fraction with a common denominator:
\[
8 = \frac{32}{4}
\]
So, we have:
\[
y = -\frac{9}{4}x - \frac{81}{4} + \frac{32}{4}
\]
Combine the constants:
\[
y = -\frac{9}{4}x - \frac{49}{4}
\]

5. **Final equation in slope-intercept form:**
Thus, the equation of the line in slope-intercept form that is perpendicular to the given line and passes through the point \((-9, 8)\) is:
\[
y = -\frac{9}{4}x - \frac{49}{4}
\]