### Video Transcript

Two cars π΄ and π΅ are moving in the same direction on a straight road, where the velocity of car π΅ relative to car π΄ is 34 kilometers per hour. If car π΄ slowed down to a quarter of its velocity, the velocity of car π΅ relative to car π΄ would be 76 kilometers per hour. Determine the actual speeds of the two cars π sub π΄ and π sub π΅.

In this question, we are dealing with relative velocities. And we know that the velocity of car π΅ relative to car π΄ is equal to the velocity of car π΅ minus the velocity of car π΄. This formula holds for any two bodies moving along the same one-dimensional axis. In this question, we are told this is equal to 34 kilometers per hour, giving us the equation π sub π΅ minus π sub π΄ is equal to 34. We are also told that if car π΄ slows down to a quarter of its velocity, the relative velocity is 76 kilometers per hour. This means that π sub π΅ minus a quarter of π sub π΄ is equal to 76. We now have a pair of simultaneous equations that we can solve to calculate π sub π΄ and π sub π΅.

We can solve these equations by elimination or substitution. In this question, we will rearrange equation one first. Adding π sub π΄ to both sides of this equation, we have π sub π΅ is equal to π sub π΄ plus 34. We can now substitute this into equation two. This gives us π sub π΄ plus 34 minus a quarter π sub π΄ is equal to 76. Collecting like terms on the left-hand side and subtracting 34 from both sides, we have three-quarters π sub π΄ is equal to 42. We can then divide through by three-quarters such that π sub π΄ is equal to 56. The speed of car π΄ is 56 kilometers per hour.

We can then substitute this value back in to equation one. We have π sub π΅ minus 56 is equal to 34. Adding 56 to both sides of this equation gives us π sub π΅ is equal to 90. The speed of car π΅ is 90 kilometers per hour. We can therefore conclude that the actual speeds of the two cars π΄ and π΅ are 56 and 90 kilometers per hour, respectively.