https://file.aiquickdraw.com/mathbot/file/1729286261521tsfk99ge.jpg

We are tasked with writing the equation of a line that is **perpendicular** to the given line equation:

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

and passes through the point \((-8, 9)\).

### Step 1: Find the slope of the perpendicular line
The slope of the given line is \(\frac{5}{6}\). For a line to be perpendicular to another, the slopes must be negative reciprocals of each other.

So, the slope \(m_{\text{perpendicular}}\) is:

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

### Step 2: Use the point-slope form of the equation
The point-slope form is:

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

where \((x_1, y_1)\) is the given point and \(m\) is the slope. Substituting \((-8, 9)\) and \(m = -\frac{6}{5}\), we get:

\[
y - 9 = -\frac{6}{5}(x + 8)
\]

### Step 3: Simplify the equation
We can leave it in point-slope form or simplify to slope-intercept form (\(y = mx + b\)).

Distribute the slope:

\[
y - 9 = -\frac{6}{5}x - \frac{6}{5} \times 8
\]

\[
y - 9 = -\frac{6}{5}x - \frac{48}{5}
\]

Now, add 9 to both sides:

\[
y = -\frac{6}{5}x - \frac{48}{5} + 9
\]

Convert 9 into a fraction with denominator 5:

\[
y = -\frac{6}{5}x - \frac{48}{5} + \frac{45}{5}
\]

Simplify:

\[
y = -\frac{6}{5}x - \frac{3}{5}
\]

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

\[
y = -\frac{6}{5}x - \frac{3}{5}
\]

Let me know if you'd like further details or have any questions.

Here are 5 related questions:
1. What is the slope of a line parallel to the given one?
2. How do you find the equation of a line passing through two given points?
3. How does the point-slope form relate to the slope-intercept form?
4. How do perpendicular lines behave in geometry?
5. What are the negative reciprocal slopes, and why do they result in perpendicular lines?

**Tip:** When finding the perpendicular slope, always remember it is the negative reciprocal of the original slope.

Write the equation of the line perpendicular to y = (5/6)x + 5/6 that passes through the point (-8, 9).

Learn how to find the equation of a line perpendicular to y = (5/6)x + 5/6 passing through point (-8, 9) using point-slope and slope-intercept forms. This covers core algebra concepts like slope relationships and perpendicular lines.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation