Give the equation of the line passing through the point  
(
15
,
−
17
)
  that is perpendicular to  
y
=
5
6
x
+
6
 .

Write your answer in slope- intercept form.

To find the equation of the line that is perpendicular to the line \( y = \frac{5}{6}x + 6 \) and passes through the point \( (15, -17) \), we can follow these steps:

### 1. Find the slope of the perpendicular line.
The slope of the given line is \( m = \frac{5}{6} \). The slope of a line perpendicular to this will be the negative reciprocal of \( \frac{5}{6} \). So, the perpendicular slope \( m_{\text{perpendicular}} \) is:

\[
m_{\text{perpendicular}} = -\frac{6}{5}
\]

### 2. Use the point-slope form of the equation.
The point-slope form of the equation of a line is:

\[
y - y_1 = m(x - x_1)
\]

Here, \( m = -\frac{6}{5} \) and the point \( (x_1, y_1) \) is \( (15, -17) \). Plugging in these values:

\[
y + 17 = -\frac{6}{5}(x - 15)
\]

### 3. Simplify to slope-intercept form.
First, distribute the slope:

\[
y + 17 = -\frac{6}{5}x + \frac{6}{5} \times 15
\]

Calculate \( \frac{6}{5} \times 15 \):

\[
\frac{6}{5} \times 15 = 18
\]

So the equation becomes:

\[
y + 17 = -\frac{6}{5}x + 18
\]

Subtract 17 from both sides to isolate \( y \):

\[
y = -\frac{6}{5}x + 1
\]

### Final Answer:
The equation of the line in slope-intercept form is:

\[
\boxed{y = -\frac{6}{5}x + 1}
\]

Would you like more details or have any questions? Here are five related questions to consider:

1. How do you find the slope of a line given two points?
2. What is the point-slope form of a line, and when is it used?
3. How do you find the equation of a line parallel to a given line?
4. Can you find the y-intercept of a line given its equation in standard form?
5. How does the slope of a line affect its angle relative to the x-axis?

**Tip:** The negative reciprocal of a slope \( m \) is \( -\frac{1}{m} \), which gives the slope of a perpendicular line.

Learn how to find the equation of a line perpendicular to y = 5/6x + 6 that passes through the point (15, -17). This problem covers the concept of slopes and perpendicular lines, with step-by-step instructions and examples. Suitable for Grades 9-12.

Math Problem Statement

Give the equation of the line passing through the point
( 15 , − 17 ) that is perpendicular to
y

Solution

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