Math Problem Statement
correct?
Solution
Let's check the correctness of the solutions:
Given Line:
The line provided is:
Step 1: Finding the Slope of the Given Line
The equation is in slope-intercept form , where represents the slope. Here, the slope of the given line is:
Step 2: Perpendicular Line
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Finding the Perpendicular Slope: The slope of a line perpendicular to the given line will be the negative reciprocal of . So, the perpendicular slope is:
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Equation of the Perpendicular Line Passing Through (3, -2): Using the point-slope form , where and , we get: Expanding this: Subtracting 2 (or ) from both sides:
The correct equation for the perpendicular line should be:
However, the image solution shows: This is incorrect. The correct y-intercept should be .
Step 3: Parallel Line
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Finding the Parallel Slope: Since the parallel line has the same slope as the given line, it will also have .
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Equation of the Parallel Line Passing Through (3, -2): Using the point-slope form with and , we get: Expanding this: Subtracting 2 from both sides:
The correct equation for the parallel line should be:
However, the image solution shows: This is also incorrect. The correct y-intercept should be .
Summary
The correct answers are:
- Equation of the perpendicular line:
- Equation of the parallel line:
Both equations in the image are incorrect. Let me know if you'd like further details on any of these steps.
Here are some related questions you might find helpful:
- How do you determine the slope of a line from its equation?
- Why is the perpendicular slope the negative reciprocal of the original slope?
- What is the point-slope form of a line, and when is it useful?
- How can you check if two lines are parallel or perpendicular using their slopes?
- Why is expanding and simplifying important when working with point-slope form?
Tip: Always double-check calculations by substituting the given point into the final equation to verify if it satisfies it.
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Math Problem Analysis
Mathematical Concepts
Algebra
Slopes of Lines
Perpendicular and Parallel Lines
Formulas
Slope-intercept form y = mx + b
Point-slope form y - y1 = m(x - x1)
Negative reciprocal slope for perpendicular lines
Theorems
Parallel and Perpendicular Slopes Theorem
Suitable Grade Level
Grades 8-10
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