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We are tasked with finding the equation of line \( h \), which is perpendicular to line \( g \) and passes through the point \( (-5, 9) \).

### Step 1: Find the slope of line \( g \)
The equation of line \( g \) is:
\[
3x - 5y = -10
\]
We need to rewrite this in slope-intercept form \( y = mx + b \) to find the slope.

Rearranging the terms:
\[
-5y = -3x - 10
\]
\[
y = \frac{3}{5}x + 2
\]
So, the slope \( m_g \) of line \( g \) is \( \frac{3}{5} \).

### Step 2: Find the slope of line \( h \)
Since line \( h \) is perpendicular to line \( g \), the slope of line \( h \), \( m_h \), will be the negative reciprocal of \( m_g \). The negative reciprocal of \( \frac{3}{5} \) is \( -\frac{5}{3} \).

### Step 3: Use the point-slope form to find the equation of line \( h \)
Now that we have the slope \( m_h = -\frac{5}{3} \) and a point \( (-5, 9) \) that lies on line \( h \), we use the point-slope form:
\[
y - y_1 = m(x - x_1)
\]
Substitute \( m = -\frac{5}{3} \), \( x_1 = -5 \), and \( y_1 = 9 \):
\[
y - 9 = -\frac{5}{3}(x + 5)
\]
Simplifying:
\[
y - 9 = -\frac{5}{3}x - \frac{25}{3}
\]
\[
y = -\frac{5}{3}x - \frac{25}{3} + 9
\]
Convert 9 to a fraction with a denominator of 3:
\[
y = -\frac{5}{3}x - \frac{25}{3} + \frac{27}{3}
\]
\[
y = -\frac{5}{3}x + \frac{2}{3}
\]

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

Would you like any further details or explanations?

Here are 5 related questions to expand on this:
1. How do you find the equation of a line parallel to a given line?
2. What is the geometric significance of the slope being the negative reciprocal?
3. How do you convert between different forms of a line equation (standard, point-slope, slope-intercept)?
4. How can you determine if two lines are perpendicular using only their equations?
5. Can you find the equation of a line parallel to line \( g \) that passes through a different point?

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

The equation of line g is 3x - 5y = -10. Line h, which is perpendicular to line g, includes the point (-5, 9). What is the equation of line h?

Discover how to find the equation of a line that is perpendicular to another line and passes through a specific point, using slope-intercept and point-slope forms. This lesson covers the essential concepts of linear algebra and coordinate geometry suitable for high school students.

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