Math Problem Statement
Solution
The question asks to find the equation of a line perpendicular to the given line and passing through the point .
Step-by-step solution:
-
Rewrite the given equation:
The line equation is . Let's rewrite it in slope-intercept form to identify its slope: So, the slope of this line is . -
Find the slope of the perpendicular line:
The slope of a line perpendicular to another is the negative reciprocal of the original line's slope.
Therefore, the slope of the new line is: -
Use the point-slope form:
The point-slope form of a line is given by: where is a point on the line, and is the slope. Using the point and the slope , we have: -
Simplify the equation:
Distribute and simplify the equation:
So, the equation of the line perpendicular to and passing through is:
Would you like more details or explanations? Here are a few related questions to consider:
- How do you convert any linear equation to slope-intercept form?
- Why is the slope of a perpendicular line the negative reciprocal?
- What are other ways to find the equation of a line besides point-slope form?
- How would you graph two perpendicular lines on the coordinate plane?
- What happens if the line is parallel instead of perpendicular?
Tip: To verify your work, always check that the slopes of perpendicular lines multiply to .
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Math Problem Analysis
Mathematical Concepts
Algebra
Linear Equations
Perpendicular Lines
Slope-Intercept Form
Formulas
Slope-Intercept Form: y = mx + b
Point-Slope Form: y - y1 = m(x - x1)
Negative Reciprocal of the Slope
Theorems
Slope of Perpendicular Lines: The product of the slopes of two perpendicular lines is -1.
Suitable Grade Level
Grades 9-10
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