Line v has an equation of 2x–3y=9. Line w, which is perpendicular to line v, includes the point (2,2). What is the equation of line w?
Write the equation in slope-intercept form. Write the numbers in the equation as simplified proper fractions, improper fractions, or integers.

To find the equation of line \( w \) that is perpendicular to line \( v \) and passes through the point \( (2, 2) \), we can follow these steps:

### Step 1: Find the slope of line \( v \)
The equation of line \( v \) is given as:
\[
2x - 3y = 9
\]
We can rewrite this equation in slope-intercept form (\( y = mx + b \)) to identify the slope (\( m \)).

Solve for \( y \):
\[
-3y = -2x + 9
\]
\[
y = \frac{2}{3}x - 3
\]
The slope of line \( v \) is \( \frac{2}{3} \).

### Step 2: Find the slope of line \( w \)
Since line \( w \) is perpendicular to line \( v \), the slope of line \( w \) will be the negative reciprocal of the slope of line \( v \).

The slope of line \( v \) is \( \frac{2}{3} \), so the negative reciprocal is:
\[
m_w = -\frac{3}{2}
\]
Thus, the slope of line \( w \) is \( -\frac{3}{2} \).

### Step 3: Use the point-slope form to find the equation of line \( w \)
Now that we have the slope of line \( w \) and a point on the line, \( (2, 2) \), we can use the point-slope form of a line's equation:
\[
y - y_1 = m(x - x_1)
\]
where \( (x_1, y_1) = (2, 2) \) and \( m = -\frac{3}{2} \).

Substitute the values:
\[
y - 2 = -\frac{3}{2}(x - 2)
\]
Distribute the slope on the right-hand side:
\[
y - 2 = -\frac{3}{2}x + 3
\]
Solve for \( y \):
\[
y = -\frac{3}{2}x + 5
\]

### Final Answer:
The equation of line \( w \) in slope-intercept form is:
\[
y = -\frac{3}{2}x + 5
\]

Would you like more details or have any questions?

Here are five related questions to further explore:
1. How do you find the equation of a line parallel to line \( v \)?
2. What are the conditions for two lines to be perpendicular?
3. How can you convert between point-slope and slope-intercept forms?
4. What is the general formula for finding the slope of a line from its equation in standard form?
5. How do you find the intersection point between two lines?

**Tip:** When working with perpendicular lines, always remember that their slopes are negative reciprocals of each other.

Line v has an equation of 2x–3y=9. Line w, which is perpendicular to line v, includes the point (2,2). What is the equation of line w? Write the equation in slope-intercept form. Write the numbers in the equation as simplified proper fractions, improper fractions, or integers.

This tutorial helps you find the equation of a line perpendicular to 2x–3y=9 that passes through the point (2,2). The solution involves key concepts like slope-intercept form, point-slope form, and the rule for perpendicular lines. Ideal for students in Grades 9-10.

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