Math Problem Statement
Use the given line and point to write an equation of a line that is parallel and perpendicular to the given equation.
Solution
Let's solve the problem step by step. You are given the line equation and the point . The task is to find:
- An equation of a line parallel to the given line that passes through the point .
- An equation of a line perpendicular to the given line that passes through the point .
Step 1: Find the slope of the given line
The given equation is in slope-intercept form , where is the slope.
The slope of the given line is .
Step 2: Equation of a line parallel to the given line
A line parallel to the given line will have the same slope, .
Using the point-slope form of a line equation:
Substitute , :
Simplify:
The equation of the parallel line is:
Step 3: Equation of a line perpendicular to the given line
The slope of a line perpendicular to another line is the negative reciprocal of the original slope.
The slope of the given line is , so the slope of the perpendicular line is:
Using the point-slope form of a line equation: Substitute , : Simplify:
The equation of the perpendicular line is:
Final Answer:
- The equation of the parallel line is .
- The equation of the perpendicular line is .
Would you like more details or additional explanation?
Related Questions:
- How can you verify that two lines are parallel or perpendicular?
- What is the general formula to find a line passing through a point with a given slope?
- How do we convert slope-intercept form to standard form?
- What happens if the given line is vertical or horizontal?
- How do we interpret the graph of these lines?
Tip:
To check if your equations are correct, substitute the given point into the equations to ensure they satisfy the line equations!
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Math Problem Analysis
Mathematical Concepts
Linear Equations
Point-Slope Form
Slope of a Line
Formulas
y - y1 = m(x - x1)
Slope of perpendicular line = -1 / (Slope of the original line)
Theorems
Parallel Line Theorem
Perpendicular Line Theorem
Suitable Grade Level
Grades 8-10
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