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Amy is helping plan her school's new basketball court. The west edge of the basketball court is located on the line [tex][tex]$y=-4 x$[/tex][/tex]. The east edge cannot intersect with the west edge. On which line could the east edge be located?

A. [tex][tex]$y - 4 x = -200$[/tex][/tex]
B. [tex][tex]$-4 x - y = -50$[/tex][/tex]
C. [tex][tex]$4 x - y = -200$[/tex][/tex]
D. [tex][tex]$-y + 4 x = -50$[/tex][/tex]

Sagot :

GinnyAnswer
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]
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