Find the equation of a line perpendicular to3, x, equals, minus, 2, plus, 2, y3x=−2+2ythat passes through the point left parenthesis, minus, 9, comma, 3, right parenthesis(−9,3).
Answer
Multiple Choice Answers
y, minus, 3, equals, minus, start fraction, 2, divided by, 3, end fraction, left parenthesis, x, plus, 9, right parenthesisy−3=− 
3
2
​
 (x+9)
y, plus, 3, equals, start fraction, 2, divided by, 3, end fraction, left parenthesis, x, plus, 9, right parenthesisy+3= 
3
2
​
 (x+9)
y, minus, 3, equals, minus, start fraction, 2, divided by, 3, end fraction, left parenthesis, x, minus, 9, right parenthesisy−3=− 
3
2
​
 (x−9)
y, minus, 3, equals, start fraction, 3, divided by, 2, end fraction, left parenthesis, x, plus, 9, right parenthesisy−3= 
2
3
​
 (x+9)

We are tasked with finding the equation of a line that is perpendicular to \( 3x = -2 + 2y \) and passes through the point \( (-9, 3) \).

### Step 1: Rewrite the given line in slope-intercept form
Start by rewriting the given equation \( 3x = -2 + 2y \):

\[
3x + 2 = 2y
\]

Now, solve for \( y \):

\[
2y = 3x + 2
\]
\[
y = \frac{3}{2}x + 1
\]

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

### Step 2: Find the slope of the perpendicular line
The slope of a line perpendicular to another line is the negative reciprocal of the original slope. Since the slope of the given line is \( \frac{3}{2} \), the slope of the perpendicular line will be:

\[
\text{slope} = -\frac{2}{3}
\]

### Step 3: Use the point-slope form
Now, use the point-slope form of the equation of a line, which is:

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

Here, \( m = -\frac{2}{3} \) and the point is \( (-9, 3) \). Substituting these values into the formula:

\[
y - 3 = -\frac{2}{3}(x + 9)
\]

### Step 4: Simplify the equation
We now have the equation of the perpendicular line:

\[
y - 3 = -\frac{2}{3}(x + 9)
\]

This matches the first choice in the multiple-choice answers:

\[
\boxed{y - 3 = -\frac{2}{3}(x + 9)}
\]

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Would you like a deeper explanation or have any further questions on this? Here are five related questions for you to explore:

1. How do you find the slope of a line given two points?
2. What is the general formula for a line perpendicular to a given line?
3. Can a line have an undefined slope? If so, what does that look like?
4. How can you graph a line using its slope and a point?
5. What is the difference between the slope-intercept and point-slope forms of a line?

**Tip:** Always remember that perpendicular lines have slopes that are negative reciprocals of each other!

Find the equation of a line perpendicular to 3x = −2 + 2y that passes through the point (−9, 3).

Learn how to find the equation of a line perpendicular to 3x = -2 + 2y that passes through the point (-9, 3). This problem involves key algebraic concepts such as linear equations, slopes, and the point-slope form, suitable for students in Grades 9-10.

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