Video Transcript
The relation 𝑆 is defined on the positive real numbers by 𝑥 𝑆 𝑦, if and only if 𝑦 squared equals four 𝑥. Determine the values of 𝑎, 𝑏, 𝑐, and 𝑑 given that 𝑎, one; nine, 𝑏; 𝑐, seven; and 121 over four, 𝑑 belong to 𝑆.
Remember, a relation is a rule between two sets that results in a collection of ordered pairs containing one object from each set. In this example, our relation 𝑆 links elements defined generally by 𝑥 in one set with those in a second set. These are defined generally as 𝑦. We are then told the specific rule that links 𝑥 and 𝑦. It’s 𝑦 squared equals four 𝑥. In other words, for a given value of 𝑥 in the first set, if we multiply that by four, we’ll get the equivalent value of 𝑦 from the second set squared.
Let’s now consider the first ordered pair. In this case, 𝑥 equals 𝑎 and 𝑦 equals one. Then, since the relation links four 𝑥 with 𝑦 squared, we calculate the value of these expressions. Four 𝑥 is four 𝑎, and 𝑦 squared is one squared, which is one. Since the equation says that 𝑦 squared is equal to four 𝑥, we form the following equation: one equals four 𝑎. To solve this equation, we divide by four, meaning that one-quarter equals 𝑎 or 𝑎 equals a quarter.
Let’s repeat this process with the second ordered pair. In this case, 𝑥 equals nine and 𝑦 equals 𝑏. Now, four 𝑥 is four times nine, which is 36, and 𝑦 squared is equal to 𝑏 squared. That gives us a new equation, 36 equals 𝑏 squared. To solve this equation for 𝑏, we take the positive and negative square root of 36. So 𝑏 equals positive or negative six. However, whilst that’s the general solution to this equation, the relation is defined only over the positive numbers. So we choose the solution 𝑏 equals positive six.
Let’s now look at the third ordered pair. This time, four 𝑥 equals four 𝑐, and 𝑦 squared equals 49. So, 49 equals four 𝑐, which we solve by dividing through by four, giving 𝑐 equals 49 over four.
Finally, for our fourth ordered pair, we have four 𝑥 equals 121 and 𝑦 squared equals 𝑑 squared. We form the equation 𝑑 squared equals 121, meaning that 𝑑 is equal to positive or negative root 121. That’s positive or negative 11. But once again, we remember that this relation is defined on the positive real numbers, so we choose positive 11.
So we have the values of 𝑎, 𝑏, 𝑐, and 𝑑. 𝑎 is one-quarter, 𝑏 is six, 𝑐 is 49 over four, and 𝑑 is 11.
