Question Video: Determining Which Ordered Pairs Satisfy a Given Linear Relation Mathematics • Second Year of Preparatory School

Video Transcript

Which of the following pairs all satisfy the relation 𝑥 minus 𝑦 equals negative 13? Option (A) negative 21, negative eight; negative 19, negative six; negative 17, negative four; eight, negative five. Option (B) negative 21, negative eight; negative 19, negative six; negative 17, negative four; negative five, eight. Option (C) negative five, negative eight; negative seven, negative six; negative nine, negative four; negative 21, eight. Option (D) negative five, negative eight; negative seven, negative six; negative nine, negative four; eight, negative 21. Or is it option (E) negative eight, negative five; negative six, negative seven; negative four, negative nine; eight, negative 21?

In this question, we are given multiple lists of pairs of numbers and asked to determine which list of pairs all satisfy a given relation. To do this, we first need to recall that these pairs are all ordered pairs. The first value in the ordered pair tells us the value of 𝑥, and the second value tells us the value of 𝑦.

For these ordered pairs to satisfy the relation, they must be valid solutions of the given equation. This means that we can check each pair by substituting the values of 𝑥 and 𝑦 into the equation to check that it is satisfied. This is equivalent to just checking that 𝑥 minus 𝑦 equals negative 13. So let’s calculate the difference in the values in each ordered pair.

For the first pair in option (A), we see that 𝑥 minus 𝑦 equals negative 21 minus negative eight. We can then calculate that this is equal to negative 13. So this ordered pair satisfies the relation. We can see that this pair is also the first pair in option (B). We can keep track of which pairs satisfy the relation with a tick.

Let’s follow the same process for the next ordered pair. We have that 𝑥 equals negative 19 and 𝑦 equals negative six. We calculate that 𝑥 minus 𝑦 is negative 19 minus negative six, which is equal to negative 13. This means that this ordered pair satisfies the relation. So we can move on to the next pair in the list. This time, we have 𝑥 equals negative 17 and 𝑦 equals negative four. We can calculate that 𝑥 minus 𝑦 equals negative 17 minus negative four, which is negative 13. Once again, this ordered pair satisfies the relation.

If we follow this process for the final pair in option (A), then we have 𝑥 equals eight and 𝑦 equals negative five. We can calculate that 𝑥 minus 𝑦 equals 13. Since the difference is 13 and not negative 13, we can conclude that this ordered pair does not satisfy the relation.

Let’s now move on to option (B). Since the first three pairs in options (A) and (B) are the same, we only need to check the final pair. We have that 𝑥 is negative five and 𝑦 is eight. This gives us that 𝑥 minus 𝑦 is negative five minus eight, which is negative 13. This means that the final pair in this list also satisfies the relation. So all of the pairs in this list satisfy the relation. Hence, the answer is option (B).

For due diligence, we can also check the pairs in the remaining options. We see that the first pair in options (C) and (D) are the same. And the value of 𝑥 is negative five and 𝑦 is negative eight. We can then calculate that 𝑥 minus 𝑦 is negative five minus negative eight, which equals three. This is not equal to negative 13, so this pair does not satisfy the relation.

Finally, we can apply this process to option (E). For the first ordered pair, we have 𝑥 equals negative eight and 𝑦 equals negative five. We can calculate that 𝑥 minus 𝑦 is negative three. This is not negative 13, so this ordered pair does not satisfy the relation.

Hence, only in option (B) do all of the ordered pairs satisfy the relation 𝑥 minus 𝑦 equals negative 13.