This week I learned system of linear equation problems with distance, speed and time applications.

Distance Formula:  distance = speed . time

Or with a visual representation:

Now that we know the formula, all we have to do is solve the questions with the information the formula gives us.

Question 1: A cyclist completes the 35 km journey from his home to the city centre in 1.3 hours. If the cyclist’s speed is 20 km/h on rough roads and 40 km/h on flat roads, how many km did the cyclist travel on a flat road?

First of all, I created a table like the photo above. Then, I wrote the total distance, speed on rough and flat roads, and total time, which are the information given in the question, in this table. Then, since the question asks about the km traveled on a flat road, one of the unknowns in this question is the km traveled on a flat road, so I wrote y in the distance part of the flat road in the table and x in the distance part of the rough road, which is another unknown.Then, using the distance formula I showed above, when I thought that the product of speed and time equals the distance, I wrote the time spent on the rough road as x/20 and the time spent on the straight road as y/40.

Now that I have filled in the table using the formula and according to what is given in the question, it is time to set up and solve the system. As seen in the table, the sum of x and y gives the total distance of 35. Therefore, one of the equations in our system is x + y = 35. And when we look at the time section in the table, the sum of x/20 and y/40 gives the total time of 1.3. so the second equation in our system is x/20 + y/40 = 1.3. And now, at this point, all we have to do is solve this system. I used the elimination method to solve it and found the result to be 18 km.

Question 2: A plane flying with the wind and the journey takes 4 hours to travel 860 km from country A to country B. On the return flight it takes 5 hours against the wind. Find the speed of the wind and the speed of the plane.

First of all, I created a table like the photo above. Then, I wrote the total distance given in the question and the time spent on the way and return in the table. Then, since the question asked me about the speed of the plane going with the wind and returning against the wind, and the speed of the wind, I said x+y to the speed of the plane going with the wind, because the wind increases the speed of the plane, and I called x-y to the speed of the plane going against the wind, that is, returning, which is slower due to the wind ( Here x represents the speed of the plane and y represents the speed of the wind). Then, using the distance formula, I established the equations 860 = (x + y) 4 and 860 = (x – y) 5, and both equations were ready for the system. Afterwards, the only thing left to do was to solve about this system. I solved it through elimination and found the speed of the plane to be 193.5 and the speed of the wind to be 21.5.