1. The speed of a passenger trin is 18mph faster than the speed of a freight train. The passenger train travels 260 miles in the same time it takes the feight train to travel 170 miles. Find the speed of each train.

Vf = X mi/h.=Velocity of freight train.

Vp = (X + 18)mi/h = Velocity of passenger train.

Eq1: Dp = (X+18)t = 260.

Eq2: Df = Xt = 170,
t = 170/X.

In Eq1, substitute 170/X for t:
(X+18)170/X = 260,
Multiply both sides by X:
170(X+18) = 260x,
170x + 3060 = 260x,
260x - 170x = 3060,
90x = 3060,
X = 34 mi/h = Speed of freight train.
X+18 = 34 + 18 = 52 = Speed of passenger train.

To find the speed of each train, we can start by defining variables for the speeds of the passenger train and the freight train.

Let's assume:
- The speed of the passenger train = x mph
- The speed of the freight train = y mph

According to the given information:
1. "The speed of a passenger train is 18 mph faster than the speed of a freight train." translates to x = y + 18.

2. "The passenger train travels 260 miles in the same time it takes the freight train to travel 170 miles." This gives us a time equation: time taken by passenger train / time taken by freight train = distance traveled by passenger train / distance traveled by freight train.
Plugging in the values: 260 / x = 170 / y

Now, we have two equations:
x = y + 18 (equation 1)
260 / x = 170 / y (equation 2)

To solve this system of equations, we can use the substitution method or elimination method. Let's use the substitution method here:

From equation 1, we express y in terms of x:
y = x - 18

Now, we substitute this value of y in equation 2:
260 / x = 170 / (x - 18)

Cross-multiplying:
260(x - 18) = 170x

Expanding and simplifying:
260x - 4680 = 170x

Bringing like terms together:
260x - 170x = 4680

Simplifying further:
90x = 4680

Dividing by 90:
x = 4680 / 90
x ≈ 52

So, the speed of the passenger train (x) is approximately 52 mph.

To find the speed of the freight train (y), we substitute the value of x back into equation 1:
y = x - 18
y = 52 - 18
y ≈ 34

Therefore, the speed of the freight train (y) is approximately 34 mph.

To recap:
- The speed of the passenger train is approximately 52 mph.
- The speed of the freight train is approximately 34 mph.