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Eric traveled to two cities on a single highway. The total distance one way was 200 miles. The distance from his original location to the first city was 40 miles less than the distance from the first city to the second city. Suppose that [tex]x[/tex] represents the distance from the original location to the first city and [tex]y[/tex] represents the distance from the first city to the second city. The following system of equations represents the given situation.

[tex]\[
\begin{aligned}
x + y & = 200 \\
x & = y - 40
\end{aligned}
\][/tex]

Which pair of coordinates represents the solution (in miles) to this system of equations?

A. [tex]$(70, 130)$[/tex]
B. [tex]$(80, 120)$[/tex]
C. [tex]$(160, 120)$[/tex]
D. [tex]$(100, 100)$[/tex]
E. [tex]$(140, 60)$[/tex]



Answer :

GinnyAnswer
Let's solve this problem step-by-step using the given system of equations. Here are the equations:

1. [tex]\( x + y = 200 \)[/tex]
2. [tex]\( x = y - 40 \)[/tex]

We need to find the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] that satisfy both equations.

Step 1: Substitute the expression for [tex]\( x \)[/tex] from the second equation into the first equation.
[tex]\[ x = y - 40 \][/tex]
[tex]\[ (y - 40) + y = 200 \][/tex]

Step 2: Simplify the equation.
[tex]\[ y - 40 + y = 200 \][/tex]
[tex]\[ 2y - 40 = 200 \][/tex]

Step 3: Move the constant term to the other side of the equation.
[tex]\[ 2y - 40 + 40 = 200 + 40 \][/tex]
[tex]\[ 2y = 240 \][/tex]

Step 4: Solve for [tex]\( y \)[/tex].
[tex]\[ y = \frac{240}{2} \][/tex]
[tex]\[ y = 120 \][/tex]

Step 5: Substitute [tex]\( y = 120 \)[/tex] back into the second equation to find [tex]\( x \)[/tex].
[tex]\[ x = y - 40 \][/tex]
[tex]\[ x = 120 - 40 \][/tex]
[tex]\[ x = 80 \][/tex]

So, the solution to the system of equations is [tex]\( x = 80 \)[/tex] and [tex]\( y = 120 \)[/tex].

Therefore, the pair of coordinates representing the solution (in miles) is [tex]\( (80, 120) \)[/tex].

The correct answer is:
B. [tex]\( (80, 120) \)[/tex]

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