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.