Three numbers add up to 216. The sum of the first two numbers is 112 and the third number is eight less than this sum. How many possible values are there for the numbers?
Let’s call our numbers 𝑎, 𝑏, and 𝑐. 𝑎, 𝑏, and 𝑐 add up to 216. The first two numbers 𝑎 and 𝑏 add up to 112. The third number is eight less than that sum, so 112 minus eight. If 𝑐 equals 112 minus eight, 𝑐 equals 104.
The first thing I wanna do is take our 𝑐 value, 104, and plug that in to our first statement. Now we have a statement that says 𝑎 plus 𝑏 plus 104 equals 216. We subtract 104 from either side. On the left, we have 𝑎 plus 𝑏. And on the right, we have 112. The two statements we have left both say the same thing. If we plug in 12 for 𝑎 plus 𝑏, we get a statement that says 112 equals 112. We could also substitute these equations to get something that says 𝑎 plus 𝑏 equals 𝑎 plus 𝑏. Because we have two true statements here, there are infinitely many values that 𝑎 and 𝑏 could be.
As long as 𝑎 and 𝑏 together add up to 112, infinitely many number of possibilities exist.