To determine the equation of the second cut, let's proceed with the following steps:
1. Understand that the second cut must be parallel to the first cut:
- The equation of the first cut is given by \( y = -\frac{1}{2} x + 3 \).
- For two lines to be parallel, their slopes must be equal. Therefore, the slope of the second cut should be the same as the slope of the first cut.
- From the equation \( y = -\frac{1}{2} x + 3 \), we see that the slope (m) is \( -\frac{1}{2} \).
2. Use the point-slope form of the line's equation for the second cut:
- The point through which the second cut passes is provided as \( (0, -8) \).
- The point-slope form of a line's equation is given by \( y = mx + b \).
3. Substitute the known slope and the coordinates of the point into the line's equation to find the y-intercept (b):
- Here, \( m = -\frac{1}{2} \), \( x = 0 \), and \( y = -8 \).
- Substitute these values into the equation \( y = mx + b \):
[tex]\[
-8 = -\frac{1}{2}(0) + b
\][/tex]
- Simplify the equation to solve for \( b \):
[tex]\[
-8 = b
\][/tex]
4. Form the equation of the second cut:
- Now we have the slope \( m = -\frac{1}{2} \) and the y-intercept \( b = -8 \).
- Substitute these values back into the line equation \( y = mx + b \):
[tex]\[
y = -\frac{1}{2} x - 8
\][/tex]
5. Check the possible answers:
- A. \( y = -\frac{1}{2} x + 8 \)
- B. \( y = 2 x - 8 \)
- C. \( y = -2 x + 8 \)
- D. \( y = -\frac{1}{2} x - 8 \)
Based on our calculations, the correct equation for Francine’s second cut is:
[tex]\[
\boxed{D. \, y = -\frac{1}{2} x - 8}
\][/tex]
