Answer:
Route y= 31 miles.
Route x= 24 miles.
Step-by-step explanation:
We have been given that each week Dan drives two routes: route x and route y.
One week he drives route x three times and route y 2 times. He drives a total of 134 miles that week. We can represent this information as:
Another week he drives route x twice and route y 5 times. He drives a total of 203 miles.We can represent this information as: ![2x+5y=203..(2)](https://tex.z-dn.net/?f=2x%2B5y%3D203..%282%29)
Upon using our given information we have formed a system of equations. Now we will solve our system of equations using substitution method.
From equation 1 we will get,
![x=\frac{134-2y}{3}](https://tex.z-dn.net/?f=x%3D%5Cfrac%7B134-2y%7D%7B3%7D)
Upon substituting this value in 2nd equation we will get,
![2(\frac{134-2y}{3})+5y=203](https://tex.z-dn.net/?f=2%28%5Cfrac%7B134-2y%7D%7B3%7D%29%2B5y%3D203)
Upon distributing 2 we will get,
![(\frac{268-4y}{3})+5y=203](https://tex.z-dn.net/?f=%28%5Cfrac%7B268-4y%7D%7B3%7D%29%2B5y%3D203)
Now we will find a common denominator for left side of our equation.
![\frac{268-4y+15y}{3}=203](https://tex.z-dn.net/?f=%5Cfrac%7B268-4y%2B15y%7D%7B3%7D%3D203)
![\frac{268+11y}{3}=203](https://tex.z-dn.net/?f=%5Cfrac%7B268%2B11y%7D%7B3%7D%3D203)
Upon multiplying both sides of our equation by 3 we will get,
![268+11y=3\times 203](https://tex.z-dn.net/?f=268%2B11y%3D3%5Ctimes%20203)
![268+11y=609](https://tex.z-dn.net/?f=268%2B11y%3D609)
![11y=609-268](https://tex.z-dn.net/?f=11y%3D609-268)
![11y=341](https://tex.z-dn.net/?f=11y%3D341)
![y=\frac{341}{11}=31](https://tex.z-dn.net/?f=y%3D%5Cfrac%7B341%7D%7B11%7D%3D31)
Therefore, the length of route y is 31 miles.
Now let us substitute y=31 in 1st equation to find the value of x.
![3x+2\times 31=134](https://tex.z-dn.net/?f=3x%2B2%5Ctimes%2031%3D134)
![3x+62=134](https://tex.z-dn.net/?f=3x%2B62%3D134)
![3x=134-62](https://tex.z-dn.net/?f=3x%3D134-62)
![3x=72](https://tex.z-dn.net/?f=3x%3D72)
![x=\frac{72}{3}=24](https://tex.z-dn.net/?f=x%3D%5Cfrac%7B72%7D%7B3%7D%3D24)
Therefore, the length of route x is 24 miles.