Certainly! Let's analyze each option and check if it intersects with the west edge line, which is given by the equation [tex]\( y = -4x \)[/tex].
### Step-by-Step Solution:
1. West Edge Equation:
[tex]\[
y = -4x
\][/tex]
### Analyzing Each Option:
#### Option 1: [tex]\( y - 4x = -200 \)[/tex]
Rearranging this equation, we can write it in slope-intercept form ( [tex]\( y = mx + b \)[/tex] ):
[tex]\[
y - 4x = -200 \implies y = 4x - 200
\][/tex]
Now we have:
- West edge: [tex]\( y = -4x \)[/tex]
- East edge: [tex]\( y = 4x - 200 \)[/tex]
To find the intersection, we set the equations equal to each other:
[tex]\[
-4x = 4x - 200
\][/tex]
[tex]\[
-8x = -200
\][/tex]
[tex]\[
x = 25
\][/tex]
Plugging [tex]\( x = 25 \)[/tex] back into the west edge equation:
[tex]\[
y = -4(25) = -100
\][/tex]
Thus, they intersect at [tex]\( (25, -100) \)[/tex].
#### Option 2: [tex]\( -4x - y = -50 \)[/tex]
Rearranging this equation:
[tex]\[
-4x - y = -50 \implies y = -4x + 50
\][/tex]
Now we have:
- West edge: [tex]\( y = -4x \)[/tex]
- East edge: [tex]\( y = -4x + 50 \)[/tex]
To find the intersection, we set the equations equal to each other:
[tex]\[
-4x = -4x + 50
\][/tex]
[tex]\[
0 = 50 \quad \text{(false statement)}
\][/tex]
Since this statement is false, it means the lines never intersect.
#### Option 3: [tex]\( 4x - y = -200 \)[/tex]
Rearranging this equation:
[tex]\[
4x - y = -200 \implies y = 4x + 200
\][/tex]
Now we have:
- West edge: [tex]\( y = -4x \)[/tex]
- East edge: [tex]\( y = 4x + 200 \)[/tex]
To find the intersection, we set the equations equal to each other:
[tex]\[
-4x = 4x + 200
\][/tex]
[tex]\[
-8x = 200
\][/tex]
[tex]\[
x = -25
\][/tex]
Plugging [tex]\( x = -25 \)[/tex] back into the west edge equation:
[tex]\[
y = -4(-25) = 100
\][/tex]
Thus, they intersect at [tex]\( (-25, 100) \)[/tex].
#### Option 4: [tex]\( -y + 4x = -50 \)[/tex]
Rearranging this equation:
[tex]\[
-y + 4x = -50 \implies y = 4x + 50
\][/tex]
Now we have:
- West edge: [tex]\( y = -4x \)[/tex]
- East edge: [tex]\( y = 4x + 50 \)[/tex]
To find the intersection, we set the equations equal to each other:
[tex]\[
-4x = 4x + 50
\][/tex]
[tex]\[
-8x = 50
\][/tex]
[tex]\[
x = -\frac{50}{8} = -6.25
\][/tex]
Plugging [tex]\( x = -6.25 \)[/tex] back into the west edge equation:
[tex]\[
y = -4(-6.25) = 25
\][/tex]
Thus, they intersect at [tex]\( (-6.25, 25) \)[/tex].
### Conclusion:
After analyzing all the options, the only line that does not intersect with the west edge line [tex]\( y = -4x \)[/tex] is:
[tex]\[
-4x - y = -50
\][/tex]
So, the east edge could be located on:
[tex]\[
\boxed{-4 x - y = -50}
\][/tex]
