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Let's break this down to find the equation of line \( t \).

### Step 1: Identify the slope of line \( s \)
The equation of line \( s \) is given as:

\[
y = -\frac{1}{9}x - 4
\]

From this equation, the slope of line \( s \) is \( m_s = -\frac{1}{9} \).

### Step 2: Find the slope of line \( t \)
Since line \( t \) is perpendicular to line \( s \), the slopes of perpendicular lines are negative reciprocals of each other. The negative reciprocal of \( -\frac{1}{9} \) is:

\[
m_t = 9
\]

So, the slope of line \( t \) is \( 9 \).

### Step 3: Use the point-slope form
Line \( t \) passes through the point \( (1, 7) \). We will use the point-slope form of the equation of a line:

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

where \( (x_1, y_1) \) is the point \( (1, 7) \), and \( m \) is the slope \( 9 \).

Substituting the values:

\[
y - 7 = 9(x - 1)
\]

### Step 4: Simplify to slope-intercept form
Now, expand and simplify the equation:

\[
y - 7 = 9x - 9
\]

\[
y = 9x - 9 + 7
\]

\[
y = 9x - 2
\]

### Final Answer:
The equation of line \( t \) is:

\[
y = 9x - 2
\]

Would you like any further details or clarification on this?

Here are five related questions:
1. How do you find the slope of a line given its equation in standard form?
2. What is the relationship between the slopes of perpendicular lines?
3. How do you convert between different forms of linear equations (slope-intercept, point-slope, standard)?
4. Can two lines with the same slope ever be perpendicular? Why or why not?
5. How do you graph a line given its slope and a point on the line?

**Tip**: Always check that the slope of the new line is the negative reciprocal when dealing with perpendicular lines.

The equation of line s is y = -1/9 x - 4. Line t includes the point (1, 7) and is perpendicular to line s. What is the equation of line t?

Learn how to find the equation of a line perpendicular to a given line passing through a specific point. The problem involves determining slopes of perpendicular lines and using point-slope form to derive the final equation in slope-intercept form. Suitable for Grades 8-10 students.

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