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:
Step 1: Convert the line into slope-intercept form (y = mx + b)
To find the slope of the given line, we first solve for :
So, the slope of the original line is:
Step 2: Equation of the perpendicular line
The slope of a line perpendicular to this will be the negative reciprocal of . So, the slope of the perpendicular line is:
Now, let the point the perpendicular line passes through be . The equation of the line in point-slope form is:
Substituting :
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., . Let the parallel line pass through a point . The equation of the parallel line in point-slope form is:
Substituting :
This is the equation of the parallel line.
Summary
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The equation of the perpendicular line is:
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The equation of the parallel line is:
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:
- What is the general form of a line equation, and how is it used?
- How do we calculate the slope of a line from two points?
- What is the significance of perpendicular slopes in geometry?
- How can parallel lines be used in geometric proofs?
- 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
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