Write an equation for a line passing through the point left parenthesis 4 comma negative 2 right parenthesis that is parallel to the line 4 x plus 7 y equals 8. Then write a second equation for a line passing through the point left parenthesis 4 comma negative 2 right parenthesis that is perpendicular to the line 4 x plus 7 y equals 8.


To find the equations of the lines that are parallel and perpendicular to the line \( 4x + 7y = 8 \) and pass through the point \( (4, -2) \), we need to follow these steps:

### Step 1: Convert the given equation to slope-intercept form
First, rewrite the equation \( 4x + 7y = 8 \) in slope-intercept form \( y = mx + b \) to determine the slope.

\[
4x + 7y = 8
\]
\[
7y = -4x + 8
\]
\[
y = -\frac{4}{7}x + \frac{8}{7}
\]

From this, we see that the slope of the given line is \( m = -\frac{4}{7} \).

### Step 2: Write the equation of the parallel line
Parallel lines have the same slope, so the line passing through \( (4, -2) \) will have the same slope \( m = -\frac{4}{7} \).

Using the point-slope form \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) = (4, -2) \):

\[
y - (-2) = -\frac{4}{7}(x - 4)
\]
\[
y + 2 = -\frac{4}{7}(x - 4)
\]

Now simplify:

\[
y + 2 = -\frac{4}{7}x + \frac{16}{7}
\]
\[
y = -\frac{4}{7}x + \frac{16}{7} - \frac{14}{7}
\]
\[
y = -\frac{4}{7}x + \frac{2}{7}
\]

Thus, the equation of the line parallel to \( 4x + 7y = 8 \) and passing through \( (4, -2) \) is:
\[
y = -\frac{4}{7}x + \frac{2}{7}
\]

### Step 3: Write the equation of the perpendicular line
Perpendicular lines have slopes that are negative reciprocals. The slope of the given line is \( m = -\frac{4}{7} \), so the slope of the perpendicular line will be \( m_{\perp} = \frac{7}{4} \).

Using the point-slope form for the perpendicular line:

\[
y - (-2) = \frac{7}{4}(x - 4)
\]
\[
y + 2 = \frac{7}{4}(x - 4)
\]

Simplify:

\[
y + 2 = \frac{7}{4}x - 7
\]
\[
y = \frac{7}{4}x - 7 - 2
\]
\[
y = \frac{7}{4}x - 9
\]

Thus, the equation of the line perpendicular to \( 4x + 7y = 8 \) and passing through \( (4, -2) \) is:
\[
y = \frac{7}{4}x - 9
\]

### Summary:
1. Equation of the parallel line: \( y = -\frac{4}{7}x + \frac{2}{7} \)
2. Equation of the perpendicular line: \( y = \frac{7}{4}x - 9 \)

Would you like further details or clarifications?

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Here are 5 related questions to further explore:
1. How can we derive the equation of a line in point-slope form?
2. What is the importance of the slope in determining parallel and perpendicular lines?
3. How do you convert between standard form and slope-intercept form?
4. How does the concept of negative reciprocals apply to perpendicular slopes?
5. Can you solve for the point of intersection of the two lines found here?

**Tip:** Always ensure you have the equation in slope-intercept form if you're determining parallel or perpendicular slopes—it makes it much easier!

Write an equation for a line passing through the point (4, -2) that is parallel to the line 4x + 7y = 8. Then write a second equation for a line passing through the point (4, -2) that is perpendicular to the line 4x + 7y = 8.

Discover how to find the equation of a line that passes through (4, -2) and is parallel to 4x + 7y = 8, as well as a perpendicular line. This step-by-step guide covers key algebraic concepts, including point-slope form and the relationship between slopes of parallel and perpendicular lines. Perfect for students in Grades 9-12.

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