Math Problem Statement
Find:the:eqation:of:a:line:that:is:perpendicular:to:the:graph:of:3x+2y=6:and:contains:the:point:\left(6,:-3\right)
Solution
We are asked to find the equation of a line that is perpendicular to the line given by the equation:
Step 1: Find the slope of the given line
First, rewrite the given equation in slope-intercept form, , where is the slope.
The slope of this line is .
Step 2: Find the slope of the perpendicular line
The slope of a line that is perpendicular to another line is the negative reciprocal of the original slope. So, the slope of the perpendicular line, , is:
Step 3: Use the point-slope form to find the equation
The point-slope form of the equation of a line is:
Here, and the given point is , so and .
Substitute these values into the point-slope form:
Step 4: Simplify the equation
Distribute the slope :
Now, subtract 3 from both sides:
Final Answer:
The equation of the line that is perpendicular to and passes through the point is:
Do you want more details on any step or have any questions?
Here are some related questions you might want to explore:
- What is the equation of a line parallel to passing through ?
- How do you derive the point-slope form of a line equation?
- Can two lines with negative reciprocal slopes ever be parallel?
- What is the general method to find the slope of a line from an equation in standard form?
- What are the conditions for two lines to be perpendicular?
Tip: Always remember, the slopes of perpendicular lines are negative reciprocals of each other!
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Math Problem Analysis
Mathematical Concepts
Algebra
Linear Equations
Perpendicular Slopes
Formulas
Slope-intercept form: y = mx + b
Point-slope form: y - y1 = m(x - x1)
Theorems
Slopes of Perpendicular Lines (Negative Reciprocal)
Suitable Grade Level
Grades 9-11