Math Problem Statement
Solution
To solve this problem, let's go through the steps required to find the equation of line , which is perpendicular to line and passes through the point .
Given Information
- Equation of line :
- Line passes through the point and is perpendicular to line .
Steps to Solution
-
Identify the Slope of Line :
- Line is in slope-intercept form , where is the slope.
- For line , the slope .
-
Determine the Slope of Line :
- Since line is perpendicular to line , its slope will be the negative reciprocal of .
- Therefore, .
-
Use the Point-Slope Form to Find the Equation of Line :
- We have the point and the slope .
- The point-slope form of a line is: Substituting , , and : Simplify the equation:
-
Simplify to Slope-Intercept Form :
- Distribute the slope on the right side:
- Calculate :
- Subtract 3 from both sides to isolate :
Final Answer
The equation of line in slope-intercept form is:
Would you like further details on any step, or do you have any questions?
Here are some related questions to expand your understanding:
- How do you find the equation of a line parallel to a given line?
- What is the significance of the negative reciprocal in perpendicular slopes?
- How would you rewrite the equation if the point given was different?
- Can you convert the equation into standard form?
- What happens to the line if you change the slope while keeping the same point?
Tip: When working with perpendicular lines, remember that their slopes are always negative reciprocals. This concept is useful in geometry and analytic geometry problems.
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Math Problem Analysis
Mathematical Concepts
Algebra
Coordinate Geometry
Slope-Intercept Form
Perpendicular Lines
Formulas
Slope-intercept form: y = mx + b
Point-slope form: y - y1 = m(x - x1)
Theorems
Perpendicular slopes theorem (negative reciprocal relationship)
Suitable Grade Level
Grade 9
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