Solving Linear equation

Question

A train has been moving for 2 hours at a uniform speed, x kmh^-1 and the next 1 hour at a uniform speed y kmj^-1. During that period of time, the train has travelled 192 km. On another journey, the train has travelled a distance of 261 km when it travels at a uniform speed, x kmh^-1 for 45 minutes and at a uniform speed of y kmh-¹ for 3 hours.

Calculate the difference between the two uniform speeds, in kmh^-1.

Answer 1

2x + y = 192

multiply by 3

6x + 3y = 576

subtract

3x/4 + 3y = 261

to eliminate one variable, y:

21x/4 = 576-261 = 315

multiply by 4/21

x = 315(4/21) = 105(4)/7 = 15(4) = 60 km per hour

y = 192-2x = 192-120 = 72 km per hour

y-x = 72-60 = 12 km per hour difference in speeds

Answer 2

A train has been moving for 2 hours at a uniform speed, x kmh^-1 and the next 1 hour at a uniform speed y kmj^-1. During that period of time, the train has travelled 192 km. On another journey, the train has travelled a distance of 261 km when it travels at a uniform speed, x kmh^-1 for 45 minutes and at a uniform speed of y kmh-¹ for 3 hours.

Calculate the difference between the two uniform speeds, in kmh^-1.

Dear Octavia

The important point here is to realize that we are talking about distances (192 km in the first case and 261 in the second case). Then, we have to multiply the speeds x and y by their corresponding traveling time for each case to get distances ( distancia = speed*Time)

Case 1: Because the train has a constant speed x during 2 hours, it travelled 2x because the speed units are in km/h y the time is 2 hours. Later, this speed changed to y (also constant) for just 1 hour, then its additional travelled distance is 1y. The total distance traveled by the trains is:

2x + 1y = 192 km 192 is the sum of the distance 2x at x speed plus 1y at speed y

Case 2: In the same way we can do the same reasoning to get

(45/60)x = 3y = 261 km. The tricky stuff here is to note that now the time of the train running at an speed x is given in minutes. Then we have to convert from minutes to hours. Because 1 hour has 60 minutes, 45 minutes is equal to (45/60) hours and can be reduced to (3/4).

At the end to resolve this problem we need to resolve a lineal system with 2 unknows x and y.

Resuming;

2x + y = 192

(3/4)x + 3y = 261

Step #1

(-3) -6x - 3y = -576 I multiplied the first equation by (-3) both sides of the equality

(+1) (3/4)x + 3y = 261 I multiplied the first equation by (+1) both sides of the equality

Step#2

-6x +(3/4)x = 261-576 I added both equations to cancel the term 3y

Step#3

(4) -24x +3x = 4(261-576)

I multiplied both sides of the equality by (4) to eliminated the fraction number

Step#4

-21x = -1260 the value of speed x is 60 km/h Note that the negative sign is cancelled

x = 60 km/h

Step#5

Using the first lineal equation (without the faction) we get the value of speed y easily

2x + y = 192

y = 192 - 2x = 192 - 2(60) = 192 - 120 = 72 km/h

y = 72 km/h

Finally, the difference between the y and x speeds is: 72 - 60 = 12 km/h