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5. A train travels two different routes: the North Route and the South Route. • On Monday, the train traveled the North Route 8 times and the South Route 10 times, traveling a total of 410 miles. • On Tuesday, the train traveled the North Route 15 times and the South Route 11 times, traveling a total of 575 miles. How long is each route?

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Answer:

North Route is 20 miles, and South Route is 25 miles.

Step-by-step explanation:

Let the length of the North route be represented by x and that of the South route be represented by y.

On Monday,

8x + 10y = 410

On Tuesday,

15x + 11y = 575

So that,

8x + 10y = 410 .............. 1

15x + 11y = 575 ......... 2

Using substitution method, from equation 1;

x = [tex]\frac{410 - 10y}{8}[/tex] ................. 3

Substitute the expression for x into equation 2,

15([tex]\frac{410 - 10y}{8}[/tex]) + 11y = 575

multiply through by 8,

6150 - 150y + 88y = 4600

6150 - 62y = 4600

6150 - 4600 = 62y

1550 = 62y

divide through by 62 to have,

y = 25

Substitute the value of y in equation 3

x = [tex]\frac{410 - 10y}{8}[/tex]

  = [tex]\frac{410 - 10(25)}{8}[/tex]

  = 20

x = 20

Thus, North Route is 20 miles, and South Route is 25 miles.

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