Math Problem Statement

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

Solution

Let's start by analyzing the given line:

7x8y=2-7x - 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 yy:

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

y=78x14(dividing both sides by 8)y = -\frac{7}{8}x - \frac{1}{4} \quad \text{(dividing both sides by } -8)

So, the slope of the original line is:

m1=78m_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 m1m_1. So, the slope of the perpendicular line is:

m2=87m_2 = \frac{8}{7}

Now, let the point the perpendicular line passes through be (x1,y1)(x_1, y_1). The equation of the line in point-slope form is:

yy1=m2(xx1)y - y_1 = m_2(x - x_1)

Substituting m2=87m_2 = \frac{8}{7}:

yy1=87(xx1)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., m1=78m_1 = -\frac{7}{8}. Let the parallel line pass through a point (x2,y2)(x_2, y_2). The equation of the parallel line in point-slope form is:

yy2=m1(xx2)y - y_2 = m_1(x - x_2)

Substituting m1=78m_1 = -\frac{7}{8}:

yy2=78(xx2)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:
    yy1=87(xx1)y - y_1 = \frac{8}{7}(x - x_1)

  • The equation of the parallel line is:
    yy2=78(xx2)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.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Slopes of Lines
Point-Slope Form
Parallel and Perpendicular Lines

Formulas

Slope-intercept form: y = mx + b
Point-slope form: y - y1 = m(x - x1)
Perpendicular slope: m2 = -1/m1

Theorems

Perpendicular slopes are negative reciprocals.
Parallel lines have the same slope.

Suitable Grade Level

Grades 9-11