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We are tasked with determining the distance from the point \( P(-1, -6) \) to the line \( y = \frac{4}{3}x - \frac{7}{3} \).

The formula to find the distance from a point \((x_1, y_1)\) to a line in the form \(Ax + By + C = 0\) is given by:

\[
d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}
\]

First, let's rewrite the given line equation \( y = \frac{4}{3}x - \frac{7}{3} \) into standard form \(Ax + By + C = 0\):

\[
\frac{4}{3}x - y - \frac{7}{3} = 0
\]

Here, \(A = \frac{4}{3}\), \(B = -1\), and \(C = -\frac{7}{3}\).

Next, we substitute the coordinates of point \(P(-1, -6)\) and the values of \(A\), \(B\), and \(C\) into the distance formula.

Let me calculate that for you.The distance from point \( P(-1, -6) \) to the line \( y = \frac{4}{3}x - \frac{7}{3} \) is approximately **1.40** units, rounded to the nearest hundredth.

Would you like more details on the steps, or do you have any questions?

Here are 5 related questions:
1. How do you find the distance between two points in a plane?
2. Can you convert any line equation into the standard form?
3. What is the geometric meaning of the distance between a point and a line?
4. How would the distance change if the point \( P \) was on the line itself?
5. What happens if the slope of the line is zero?

**Tip:** It's essential to rewrite the equation of a line into standard form \(Ax + By + C = 0\) before using the distance formula.

Determine the distance of point P(-1, -6) to line A: y = (4/3)x - (7/3), rounded to the nearest hundredth.

Learn how to calculate the distance between a point P(-1, -6) and the line A: y = (4/3)x - (7/3). This problem focuses on key geometric concepts and the distance formula. Suitable for students in Grades 9-12.

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