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To solve these problems, we need to calculate the perpendicular distance from a point \( A(x_1, y_1) \) to a line in the form \( y = mx + b \). The formula for the perpendicular distance \( d \) from a point \((x_1, y_1)\) to a line \( ax + by + c = 0 \) is:

\[
d = \frac{|ax_1 + by_1 + c|}{\sqrt{a^2 + b^2}}
\]

To apply this formula, we’ll rearrange each line equation into the standard form \( ax + by + c = 0 \).

### Problem 7

- **Point**: \( A(-4, 4) \)
- **Line**: \( y = 0.8x - 0.4 \)

Rewrite the line as:
\[
0.8x - y - 0.4 = 0
\]
So, \( a = 0.8 \), \( b = -1 \), and \( c = -0.4 \).

Now, substitute \( x_1 = -4 \), \( y_1 = 4 \) into the distance formula:
\[
d = \frac{|0.8(-4) - 1(4) - 0.4|}{\sqrt{(0.8)^2 + (-1)^2}}
\]

### Problem 8

- **Point**: \( A(-3, -3) \)
- **Line**: \( y = 0.5x + 6.5 \)

Rewrite the line as:
\[
0.5x - y + 6.5 = 0
\]
So, \( a = 0.5 \), \( b = -1 \), and \( c = 6.5 \).

Now, substitute \( x_1 = -3 \), \( y_1 = -3 \) into the distance formula:
\[
d = \frac{|0.5(-3) - 1(-3) + 6.5|}{\sqrt{(0.5)^2 + (-1)^2}}
\]

I’ll calculate each distance.The distances from each point to the corresponding line are:

1. **Problem 7**: The distance from point \( A(-4, 4) \) to the line \( y = 0.8x - 0.4 \) is approximately **5.9** units.
2. **Problem 8**: The distance from point \( A(-3, -3) \) to the line \( y = 0.5x + 6.5 \) is approximately **7.2** units.

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

### Related Questions:
1. How do you find the distance from a point to a vertical or horizontal line?
2. How can the formula be derived geometrically?
3. What are practical applications of point-to-line distance calculations?
4. How does this formula change in three-dimensional space?
5. How do you determine if a point lies on a line?

### Tip:
Always rewrite a line equation in the form \( ax + by + c = 0 \) to apply the point-to-line distance formula easily.

Find the distance from point A to the given line in each problem:

1. A(-4, 4), y = 0.8x - 0.4
2. A(-3, -3), y = 0.5x + 6.5

Learn how to find the perpendicular distance from a point to a given line using the point-to-line distance formula. This example includes solving for points A(-4, 4) to y = 0.8x - 0.4 and A(-3, -3) to y = 0.5x + 6.5.

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