Consider the line y=-9x+4


Find the equation of the line that is parallel to this line and passes through the point (-5,5)

.

Find the equation of the line that is perpendicular to this line and passes through the point (-5,5)

.

### 1. Equation of the Parallel Line

The slope of the given line \( y = -9x + 4 \) is \(-9\). Parallel lines have the same slope, so the slope of the parallel line is also \(-9\).

Using the point-slope form:

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

Substitute \((x_1, y_1) = (-5, 5)\) and \(m = -9\):

\[
y - 5 = -9(x + 5)
\]

Simplifying:

\[
y - 5 = -9x - 45
\]

\[
y = -9x - 40
\]

So, the equation of the line parallel to \(y = -9x + 4\) and passing through \((-5, 5)\) is:

\[
\boxed{y = -9x - 40}
\]

### 2. Equation of the Perpendicular Line

For the perpendicular line, the slope will be the negative reciprocal of \(-9\). The negative reciprocal of \(-9\) is \(\frac{1}{9}\).

Using the point-slope form again:

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

Substitute \((x_1, y_1) = (-5, 5)\) and \(m = \frac{1}{9}\):

\[
y - 5 = \frac{1}{9}(x + 5)
\]

Simplifying:

\[
y - 5 = \frac{1}{9}x + \frac{5}{9}
\]

\[
y = \frac{1}{9}x + \frac{5}{9} + \frac{45}{9}
\]

\[
y = \frac{1}{9}x + \frac{50}{9}
\]

So, the equation of the line perpendicular to \(y = -9x + 4\) and passing through \((-5, 5)\) is:

\[
\boxed{y = \frac{1}{9}x + \frac{50}{9}}
\]

Would you like more details on these calculations or related questions?

Here are five related questions:
1. How can you find the x-intercept of the perpendicular line you just found?
2. What is the slope-intercept form for the line perpendicular to \(y = -9x + 4\)?
3. How do you find the distance between the two parallel lines?
4. How can you determine if two lines are perpendicular just by looking at their equations?
5. What is the significance of the negative reciprocal in determining perpendicular slopes?

**Tip:** Remember, parallel lines share the same slope, while perpendicular lines have slopes that are negative reciprocals of each other.

Explore how to find equations of lines parallel and perpendicular to y = -9x + 4 passing through the point (-5, 5). Detailed step-by-step solutions provided with applications of the point-slope form. Suitable for high school students learning about linear equations and slopes.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation