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.