Find\:the\:eqation\:of\:a\:line\:that\:is\:perpendicular\:to\:the\:graph\:of\:3x+2y=6\:and\:contains\:the\:point\:\left(6,\:-3\right)

We are asked to find the equation of a line that is **perpendicular** to the line given by the equation:

\[
3x + 2y = 6
\]

### Step 1: Find the slope of the given line

First, rewrite the given equation \( 3x + 2y = 6 \) in slope-intercept form, \( y = mx + b \), where \( m \) is the slope.

\[
2y = -3x + 6
\]

\[
y = -\frac{3}{2}x + 3
\]

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

### 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, \( m_2 \), is:

\[
m_2 = \frac{2}{3}
\]

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

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

\[
y - y_1 = m(x - x_1)
\]

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

Substitute these values into the point-slope form:

\[
y - (-3) = \frac{2}{3}(x - 6)
\]

\[
y + 3 = \frac{2}{3}(x - 6)
\]

### Step 4: Simplify the equation

Distribute the slope \( \frac{2}{3} \):

\[
y + 3 = \frac{2}{3}x - 4
\]

Now, subtract 3 from both sides:

\[
y = \frac{2}{3}x - 7
\]

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

\[
y = \frac{2}{3}x - 7
\]

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 \( 3x + 2y = 6 \) passing through \( (6, -3) \)?
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!

Find the equation of a line that is perpendicular to the graph of 3x + 2y = 6 and contains the point (6, -3).

Learn how to find the equation of a line perpendicular to 3x + 2y = 6 that passes through the point (6, -3). This problem explores the concept of perpendicular slopes, slope-intercept form, and point-slope form, ideal for students in Grades 9-11.

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