write the equation of the line that passes through the point (3,3) and is perpendicular to a line that passes through the points (-4,7) and (-2,6)

To find the equation of the line that passes through the point \((3,3)\) and is perpendicular to a line passing through the points \((-4,7)\) and \((-2,6)\), we can follow these steps:

### Step 1: Determine the slope of the given line
First, we need to find the slope of the line passing through the points \((-4,7)\) and \((-2,6)\). The formula for the slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Substituting the given points:

\[
m = \frac{6 - 7}{-2 - (-4)} = \frac{-1}{2} = -\frac{1}{2}
\]

So, the slope of the given line is \(-\frac{1}{2}\).

### 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. Therefore, the slope of the line we want to find is:

\[
m_{\text{perpendicular}} = -\left(\frac{1}{-\frac{1}{2}}\right) = 2
\]

### Step 3: Use the point-slope form to find the equation of the line
We can now use the point-slope form of the equation of a line, which is:

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

Where \((x_1, y_1)\) is the point \((3,3)\) and \(m = 2\).

Substituting the values:

\[
y - 3 = 2(x - 3)
\]

### Step 4: Simplify to the slope-intercept form (optional)
To get the equation in slope-intercept form \(y = mx + b\), distribute and simplify:

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

Add 3 to both sides:

\[
y = 2x - 3
\]

So, the equation of the line is:

\[
\boxed{y = 2x - 3}
\]

Would you like more details or have any questions?

Here are some related questions you might find useful:
1. How do you determine the slope of a line given two points?
2. What is the relationship between the slopes of perpendicular lines?
3. How do you use the point-slope form to write the equation of a line?
4. What does it mean for two lines to be perpendicular?
5. How do you find the distance between a point and a line?

**Tip:** When dealing with perpendicular lines, always remember that their slopes multiply to \(-1\).

Learn how to find the equation of a line that passes through the point (3,3) and is perpendicular to a line passing through the points (-4,7) and (-2,6). This problem involves calculating slopes, understanding perpendicular lines, and using point-slope form. Suitable for high school students.

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