Math Problem Statement
Solution
The problem asks for two equations of lines:
- Equation of a line parallel to the given line that passes through the point .
- Equation of a line perpendicular to the same given line, passing through the same point.
Step 1: Find the equation of the parallel line.
The slope of the given line is . Since parallel lines have the same slope, the equation of the new line will also have a slope of .
We will use the point-slope form of the equation of a line: where and the point .
Substituting: Now simplify: Convert 5 to a fraction with denominator 7:
So, the equation of the parallel line is:
Step 2: Find the equation of the perpendicular line.
For a line perpendicular to , the slope will be the negative reciprocal of . The negative reciprocal of is .
Using the same point-slope form with and the point , we have: Simplify:
So, the equation of the perpendicular line is:
Final Answer:
- The equation of the line parallel to is .
- The equation of the line perpendicular to is .
Would you like more details or have any questions?
5 Related Questions:
- How do you find the slope of a line from an equation in standard form?
- What is the relationship between the slopes of parallel and perpendicular lines?
- How do you convert from point-slope form to slope-intercept form?
- What is the general form of a linear equation?
- How do you graph a line given its equation in slope-intercept form?
Tip:
When working with fractions in equations, it helps to clear denominators early if possible to simplify calculations!
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Math Problem Analysis
Mathematical Concepts
Linear Equations
Slope-Intercept Form
Parallel and Perpendicular Lines
Formulas
Point-Slope Form: y - y1 = m(x - x1)
Slope-Intercept Form: y = mx + b
Theorems
Parallel lines have the same slope.
Perpendicular lines have slopes that are negative reciprocals.
Suitable Grade Level
Grades 9-10
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