Question

Josh Wilson is traveling to California from South Carolina for summer vacation. The states are approximately 2500 miles apart driving cross-country. He stops at his parent's house in Oldham County, Texas, which is (12x 106) miles away from South Carolina and (5x + 685) miles away from California. Determine if Josh's parent's house in Texas the midpoint between South Carolina and California? Justify your answer.

Answer 1

Yes, Josh's parent's house in Texas is the midpoint between South Carolina and California

Explanation:

Let S represents South carolina C represents California and T represents Texas,

Given,

ST = 12x - 106,

TC = 5x + 685,

Also, SC = 2500 miles,

Suppose T is the midpoint of S and C,

⇒ ST = TC

12x - 106 = 5x + 685

7x = 791

⇒ x =
`(791)/(7)` = 113

So, ST =
`12(113)+106 =1356 - 106 = 1250`

Now,

`2* ST = 2* 1250=2500`

⇒ 2 ST = SC

Hence, T is the midpoint of S and C.

Answer 2

Given, distance between South Carolina and Texas =
`12x-106`

Distance between Texas and California =
`5x+685`

Let's assume, both the distance is same.

`12x-106=5x+685`

=>
`12x-5x = 685+106`

=>
`7x = 791`

=>
`x=113`

Plug x = 113 in
`5x+685`

`5(113)+685 = 1250` Miles.

Similarly,
`12(113)-106=1250` miles

Since 2500 divided by 2 = 1250 miles, hence our assumption was correct, Texas lies between both the cities.