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

A. [tex]\(y - 4x = -200\)[/tex]

B. [tex]\(-4x - y = -50\)[/tex]

C. [tex]\(4x - y = -200\)[/tex]

D. [tex]\(-y + 4x = -50\)[/tex]

Sagot :

GinnyAnswer
To determine the possible location of the east edge of the basketball court, we need to ensure it does not intersect with the west edge line, given by the equation [tex]\( y = -4x \)[/tex].

Here are the steps to determine the line equations that do not intersect with the west edge:

1. First, rewrite each given line equation into the slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope.

Equation 1: [tex]\( y - 4x = -200 \)[/tex]
- Rearrange to slope-intercept form:
[tex]\( y = 4x - 200 \)[/tex]
- The slope of this line is 4.

Equation 2: [tex]\( -4x - y = -50 \)[/tex]
- Rearrange to slope-intercept form:
[tex]\( -y = 4x - 50 \)[/tex]
[tex]\( y = -4x + 50 \)[/tex]
- The slope of this line is -4.

Equation 3: [tex]\( 4x - y = -200 \)[/tex]
- Rearrange to slope-intercept form:
[tex]\( -y = -4x - 200 \)[/tex]
[tex]\( y = 4x + 200 \)[/tex]
- The slope of this line is 4.

Equation 4: [tex]\( -y + 4x = -50 \)[/tex]
- Rearrange to slope-intercept form:
[tex]\( -y = -4x + 50 \)[/tex]
[tex]\( y = 4x - 50 \)[/tex]
- The slope of this line is 4.

2. Determine the intersections:
- The west edge has a slope of -4 (from the equation [tex]\( y = -4x \)[/tex]).
- Therefore, any line with a slope of -4 will be parallel and cannot intersect the west edge. The candidate for this is Equation 2 [tex]\( y = -4x + 50 \)[/tex], so it should be eliminated.

Lines with slopes different from -4 can intersect the west edge in a defined space but not be parallel to it. Therefore, they are eligible for the east edge location.

Given the above deductions, the lines where the east edge could be located are:

- [tex]\( y - 4x = -200 \)[/tex]
- [tex]\( 4x - y = -200 \)[/tex]
- [tex]\( -y + 4x = -50 \)[/tex]

Thus, the east edge of the basketball court could be located on one of the following lines:
- [tex]\( y - 4x = -200 \)[/tex]
- [tex]\( 4x - y = -200 \)[/tex]
- [tex]\( -y + 4x = -50 \)[/tex]
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