If 𝑦 varies directly with 𝑥 and 𝑦 equals 14 when 𝑥 equals six, determine the constant of proportionality.
A direct variation is a relationship in which the ratio of 𝑦 to 𝑥 is a constant 𝑘. We say 𝑦 varies directly with 𝑥, so this constant of proportionality that we need to find is 𝑘.
And it says to find it when 𝑦 is equal to 14 and 𝑥 is equal to six, so let’s put those in. So we replaced 𝑦 with 14 and 𝑥 with six. So to solve for 𝑘, we need to divide both sides by six. The sixes cancel on the right-hand side.
Fourteen-sixths is equal to 𝑘, but fourteen-sixths reduces. Since two goes into 14 and six, 𝑘 will be equal to seven-thirds, because two goes into 14 seven times and two goes into six three times.