Video Transcript
Given that 𝑦 is proportional to 𝑥 plus seven, where 𝑥 equals 13 when 𝑦 equals 34, find the relation between 𝑥 and 𝑦.
Now we read this as 𝑦 is proportional to 𝑥 plus seven. This symbol that looks a lot like the Greek letter for 𝛼 tells us that two objects or variables are in direct proportion to one another. And consider a pair of variables 𝑎 and 𝑏. If 𝑎 is directly proportional to 𝑏, we say that the ratio of these variables is some constant. 𝑎 divided by 𝑏 equals 𝑘, where 𝑘 is said to be the constant of variation or the constant of proportionality. This means that the ratio of 𝑦 to 𝑥 plus seven must be some constant. That is, 𝑦 divided by 𝑥 plus seven equals 𝑘.
Now in fact, we’re also told that when 𝑥 equals 13, 𝑦 equals 34. So we can substitute these values into our equation, and that will allow us to find the value of our constant 𝑘. When we do, we get 34 over 13 plus seven equals 𝑘, or 34 over 20 equals 𝑘. And we now observe that we can divide both the numerator and denominator of this fraction by two. So 𝑘 is equal to seventeen-tenths. Since this value of 𝑘 remains unchanged for all values of 𝑥 and 𝑦 with this relation, we can replace 𝑘 with seventeen-tenths in our earlier equation.
When we do, we find that 𝑦 over 𝑥 plus seven equals seventeen-tenths. And in fact, if we’re finding a relation between 𝑥 and 𝑦, we tend to make 𝑦 the subject. So let’s multiply both sides of this equation by 𝑥 plus seven. On the left-hand side, we’re simply left with 𝑦 and on the right, seventeen-tenths times 𝑥 plus seven in parentheses. And so we have the relation between 𝑥 and 𝑦. It’s 𝑦 equals seventeen-tenths times 𝑥 plus seven.
