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:

3x+2y=63x + 2y = 6

Step 1: Find the slope of the given line

First, rewrite the given equation 3x+2y=63x + 2y = 6 in slope-intercept form, y=mx+by = mx + b, where mm is the slope.

2y=3x+62y = -3x + 6

y=32x+3y = -\frac{3}{2}x + 3

The slope of this line is m1=32m_1 = -\frac{3}{2}.

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, m2m_2, is:

m2=23m_2 = \frac{2}{3}

Step 3: Use the point-slope form to find the equation

The point-slope form of the equation of a line is:

yy1=m(xx1)y - y_1 = m(x - x_1)

Here, m=23m = \frac{2}{3} and the given point is (6,3)(6, -3), so x1=6x_1 = 6 and y1=3y_1 = -3.

Substitute these values into the point-slope form:

y(3)=23(x6)y - (-3) = \frac{2}{3}(x - 6)

y+3=23(x6)y + 3 = \frac{2}{3}(x - 6)

Step 4: Simplify the equation

Distribute the slope 23\frac{2}{3}:

y+3=23x4y + 3 = \frac{2}{3}x - 4

Now, subtract 3 from both sides:

y=23x7y = \frac{2}{3}x - 7

Final Answer:

The equation of the line that is perpendicular to 3x+2y=63x + 2y = 6 and passes through the point (6,3)(6, -3) is:

y=23x7y = \frac{2}{3}x - 7

Do you want more details on any step or have any questions?

Here are some related questions you might want to explore:

  1. What is the equation of a line parallel to 3x+2y=63x + 2y = 6 passing through (6,3)(6, -3)?
  2. How do you derive the point-slope form of a line equation?
  3. Can two lines with negative reciprocal slopes ever be parallel?
  4. What is the general method to find the slope of a line from an equation in standard form?
  5. 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