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To find the equation of line \( t \), which is perpendicular to line \( s \) and passes through the point \( (-2, 9) \), follow these steps:

### Step 1: Determine the Slope of Line \( t \)
1. The equation of line \( s \) is given as:
   \[
   y = \frac{1}{3}x - 5
   \]
   This is in slope-intercept form \( y = mx + b \), where \( m = \frac{1}{3} \) is the slope of line \( s \).

2. For two lines to be perpendicular, the slopes must be negative reciprocals of each other. Therefore, the slope of line \( t \) will be:
   \[
   m_t = -3
   \]

### Step 2: Use the Point-Slope Form to Write the Equation of Line \( t \)
Now that we know the slope of line \( t \) is \( -3 \) and it passes through the point \( (-2, 9) \), we can use the point-slope form of a line:
\[
y - y_1 = m(x - x_1)
\]
Substitute \( m = -3 \), \( x_1 = -2 \), and \( y_1 = 9 \):
\[
y - 9 = -3(x + 2)
\]

### Step 3: Simplify to Slope-Intercept Form
Expand and simplify to get the equation in slope-intercept form:
\[
y - 9 = -3x - 6
\]
\[
y = -3x + 3
\]

### Final Answer
The equation of line \( t \) in slope-intercept form is:
\[
y = -3x + 3
\]

Would you like more details on any step or have any questions?

Here are some related questions you might find helpful:
1. How do you find the negative reciprocal of a fraction?
2. What is the general form of a line’s equation?
3. How do you convert from point-slope form to slope-intercept form?
4. How can you verify that two lines are perpendicular?
5. Why is the slope-intercept form useful in graphing lines?

**Tip:** When given a line and asked to find a perpendicular line, remember to find the negative reciprocal of the original slope for the perpendicular slope.

Line s has an equation of y = 1/3x - 5. Line t is perpendicular to line s and passes through (-2, 9). What is the equation of line t?

Learn how to find the equation of a line perpendicular to y = 1/3x - 5 and passing through the point (-2, 9). This problem involves using the slope-intercept form, point-slope form, and the concept of perpendicular slopes. Suitable for students in Grades 9-10.

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