https://file.aiquickdraw.com/mathbot/file/17268896446005ekzyayu.jpg

Let's break down and solve the problem shown in the image step by step.

### Step 1: Understand the equation of line \( u \)
The equation of line \( u \) is given as:
\[
y + 6 = \frac{3}{2}(x - 6)
\]
We will convert this into slope-intercept form, \( y = mx + b \), where \( m \) is the slope.

First, expand the right-hand side:
\[
y + 6 = \frac{3}{2}(x - 6) = \frac{3}{2}x - \frac{3}{2} \cdot 6 = \frac{3}{2}x - 9
\]
Next, subtract 6 from both sides:
\[
y = \frac{3}{2}x - 9 - 6 = \frac{3}{2}x - 15
\]
So, the equation of line \( u \) in slope-intercept form is:
\[
y = \frac{3}{2}x - 15
\]
The slope \( m_u \) of line \( u \) is \( \frac{3}{2} \).

### Step 2: Find the slope of line \( v \)
Line \( v \) is perpendicular to line \( u \). The slopes of two perpendicular lines are negative reciprocals of each other. Therefore, the slope \( m_v \) of line \( v \) will be the negative reciprocal of \( \frac{3}{2} \), which is:
\[
m_v = -\frac{2}{3}
\]

### Step 3: Use the point to find the equation of line \( v \)
We know that line \( v \) passes through the point \( (2, -1) \) and has a slope of \( -\frac{2}{3} \). We can now use the point-slope form of the equation:
\[
y - y_1 = m(x - x_1)
\]
Substitute \( (x_1, y_1) = (2, -1) \) and \( m = -\frac{2}{3} \):
\[
y - (-1) = -\frac{2}{3}(x - 2)
\]
Simplify:
\[
y + 1 = -\frac{2}{3}(x - 2)
\]
Now distribute the slope:
\[
y + 1 = -\frac{2}{3}x + \frac{4}{3}
\]
Subtract 1 from both sides to get it into slope-intercept form:
\[
y = -\frac{2}{3}x + \frac{4}{3} - 1
\]
Convert 1 to a fraction:
\[
y = -\frac{2}{3}x + \frac{4}{3} - \frac{3}{3} = -\frac{2}{3}x + \frac{1}{3}
\]

### Final Answer:
The equation of line \( v \) in slope-intercept form is:
\[
y = -\frac{2}{3}x + \frac{1}{3}
\]

Would you like further details or any clarifications?

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Here are 5 related questions you could explore:
1. How do you find the equation of a line given two points?
2. What is the relationship between the slopes of parallel lines?
3. How do you determine if two lines are perpendicular just by looking at their equations?
4. How would the equation of line \( v \) change if it passed through a different point?
5. How do you convert an equation from point-slope form to slope-intercept form?

**Tip**: Always remember that perpendicular slopes are negative reciprocals of each other.

Line u has an equation of y + 6 = (3/2)(x - 6). Line v is perpendicular to line u and passes through (2, -1). What is the equation of line v?

Find the equation of a line perpendicular to y + 6 = (3/2)(x - 6) that passes through the point (2, -1). This detailed step-by-step solution explores the use of slope-intercept form and the relationship between perpendicular slopes. Ideal for students in Grades 9-11.

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