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Given: A⁢B⟷ ∥ C⁢D⟷.

   If the coordinates of point *A* are (8 , 0) and the coordinates of point *B* are (3 , 7), the *y-*intercept of A⁢B⟷ is . If the coordinates of point *D* are (5 , 5), the equation of line C⁢D⟷ is *y* = *x* + .


Let's break down the problem into two parts.

### Part 1: Finding the \( y \)-intercept of line \( AB \)

The slope of the line \( AB \) can be calculated using the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Here, the coordinates of point \( A \) are \( (8, 0) \) and the coordinates of point \( B \) are \( (3, 7) \).

Substituting the values:
\[
m = \frac{7 - 0}{3 - 8} = \frac{7}{-5} = -\frac{7}{5}
\]

The equation of the line in slope-intercept form \( y = mx + b \) can be written as:
\[
y = -\frac{7}{5}x + b
\]
To find the \( y \)-intercept \( b \), substitute the coordinates of point \( A \) (since \( y = 0 \) when \( x = 8 \)):
\[
0 = -\frac{7}{5}(8) + b
\]
\[
0 = -\frac{56}{5} + b
\]
\[
b = \frac{56}{5}
\]

So, the \( y \)-intercept of line \( AB \) is \( \frac{56}{5} \).

### Part 2: Finding the equation of line \( CD \)

Since \( AB \) is parallel to \( CD \), they have the same slope. Therefore, the slope of \( CD \) is also \( -\frac{7}{5} \).

The equation of line \( CD \) can be written as:
\[
y = -\frac{7}{5}x + c
\]
We use the coordinates of point \( D \) (5, 5) to find \( c \):
\[
5 = -\frac{7}{5}(5) + c
\]
\[
5 = -7 + c
\]
\[
c = 12
\]

Thus, the equation of line \( CD \) is \( y = -\frac{7}{5}x + 12 \).

### Final Answers:
- The \( y \)-intercept of line \( AB \) is \( \frac{56}{5} \).
- The equation of line \( CD \) is \( y = -\frac{7}{5}x + 12 \), which corresponds to \( y = x + 12 \).

If there are any questions or if you'd like more details, feel free to ask! Here are some related questions:

1. How do you find the equation of a line given two points?
2. What does it mean for two lines to be parallel?
3. How can you determine if two lines are perpendicular?
4. What is the significance of the slope in the equation of a line?
5. How would you find the intersection point of two lines?

**Tip:** When dealing with equations of lines, always double-check your slope calculation to ensure accuracy.

Learn how to find the y-intercept of line AB and derive the equation of line CD using coordinate geometry. This problem involves calculating slopes, using slope-intercept form, and applying these concepts to real-world coordinates.

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