### Video Transcript

The sum of the weights of three people is 154 kilograms. If the first person is two kilograms heavier than the second person and the second person is one kilogram lighter than the third person, find each of their weights.

In this question, we are given some information about the weights of three people in kilograms. We want to find the individual weights of each of the people. To do this, we are told that the sum of the weights of three people is 154 kilograms. And we are given comparisons in the weights between two pairs of the three people.

To find these weights, we can start by saying that the weight of each person is unknown. Letβs say that the first person weighs π€ sub one kilograms. The second person weighs π€ sub two kilograms. And the third person weighs π€ sub three kilograms. We know that the sum of the weights is 154 kilograms, so we obtain the following equation.

We can also construct equations using the given comparisons in the weights. Since both comparisons involve the second person, we will write the weights of the other two people in terms of the weight of the second person. First, we are told that the first person is two kilograms heavier than the second person. This means if we add two kilograms onto the weight of the second person, their weight must be equal to that of the first person. So, π€ sub one is equal to π€ sub two plus two.

Similarly, we are told that the second person is one kilogram lighter than the third person. This means that we would need to add one kilogram onto the weight of the second person to make them have an equal weight to the third person. Hence, π€ sub three is equal to π€ sub two plus one.

We can now substitute these expressions for the weights of the first and third person into our equation for the sum of their weights to obtain an equation in terms of only the weight of the second person. This gives us that π€ sub two plus two plus π€ sub two plus π€ sub two plus one is equal to 154.

Now that we have an equation in one unknown, we can solve the equation. We start by collecting like terms on the left-hand side of the equation to get three π€ sub two plus three is equal to 154. Next, we subtract three from both sides of the equation to obtain that three π€ sub two is equal to 151. Then, we divide both sides of the equation through by three to find that π€ sub two is equal to 151 over three. Therefore, the second person weighs 151 over three kilograms.

We can use our equations for the weights of the other two people and the weight of the second person to determine the weights of all three people. We can add two onto our value of π€ sub two to get that π€ sub one is 157 over three. And we can add one onto our value of π€ sub two to find that π€ sub three is 154 over three.

We can check that these values match the information by checking that their sum is 154 kilograms and that the two comparisons are met. If we did this, we would confirm that these are the correct weights. Hence, the weight of the first person is 157 over three kilograms. The weight of the second person is 151 over three kilograms. And the weight of the third person is 154 over three kilograms.