Consider the line . -7x - 8y = 2
Find the equation of the line that is perpendicular to this line and passes through the point .
Find the equation of the line that is parallel to this line and passes through the point .
Note that the ALEKS graphing calculator may be helpful in checking your answer.
equation of perpendicular line
equation of parallel line

Let's start by analyzing the given line:

\[
-7x - 8y = 2
\]

### Step 1: Convert the line into slope-intercept form (y = mx + b)
To find the slope of the given line, we first solve for \(y\):

\[
-8y = 7x + 2 \quad \text{(adding } 7x \text{ to both sides)}
\]

\[
y = -\frac{7}{8}x - \frac{1}{4} \quad \text{(dividing both sides by } -8)
\]

So, the slope of the original line is:

\[
m_1 = -\frac{7}{8}
\]

### Step 2: Equation of the perpendicular line

The slope of a line perpendicular to this will be the negative reciprocal of \(m_1\). So, the slope of the perpendicular line is:

\[
m_2 = \frac{8}{7}
\]

Now, let the point the perpendicular line passes through be \((x_1, y_1)\). The equation of the line in point-slope form is:

\[
y - y_1 = m_2(x - x_1)
\]

Substituting \(m_2 = \frac{8}{7}\):

\[
y - y_1 = \frac{8}{7}(x - x_1)
\]

This is the equation of the perpendicular line.

### Step 3: Equation of the parallel line

The slope of the parallel line will be the same as the slope of the given line, i.e., \(m_1 = -\frac{7}{8}\). Let the parallel line pass through a point \((x_2, y_2)\). The equation of the parallel line in point-slope form is:

\[
y - y_2 = m_1(x - x_2)
\]

Substituting \(m_1 = -\frac{7}{8}\):

\[
y - y_2 = -\frac{7}{8}(x - x_2)
\]

This is the equation of the parallel line.

### Summary
- The equation of the **perpendicular line** is:  
  \[
  y - y_1 = \frac{8}{7}(x - x_1)
  \]

- The equation of the **parallel line** is:  
  \[
  y - y_2 = -\frac{7}{8}(x - x_2)
  \]

If you provide the coordinates of the points, I can give the exact equations. Would you like to proceed with that?

---

Would you like more details or have any questions?  
Here are 5 related questions:
1. What is the general form of a line equation, and how is it used?
2. How do we calculate the slope of a line from two points?
3. What is the significance of perpendicular slopes in geometry?
4. How can parallel lines be used in geometric proofs?
5. How do perpendicular and parallel line slopes help in coordinate geometry problems?

**Tip:** For perpendicular lines, always remember the slopes are negative reciprocals.

Consider the line -7x - 8y = 2. Find the equation of the line that is perpendicular to this line and passes through the point. Find the equation of the line that is parallel to this line and passes through the point. Note that the ALEKS graphing calculator may be helpful in checking your answer. Equation of perpendicular line, equation of parallel line.

Learn how to find the equations of a perpendicular and a parallel line to -7x - 8y = 2. The problem involves converting the line to slope-intercept form, using point-slope equations, and understanding the relationship between slopes for parallel and perpendicular lines. Suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation