When we talk about a ‘solution’ or a ‘solution set’ in algebra we usually mean, the real number or numbers that were described as variables (letters) in the original equation. When there is more than one correct answer to an equation, then we describe the group of all the right answers as part of the set.
The biggest difference, other than the solutions, is the way we symbolize it. A set of answers is symbolized using a capital letter in algebra. Just as little x & y are the most common symbols for variables in algebraic equations [in the English speaking world that is] capital X & Y indicate solution sets.
Let’s say that a man named John goes to the State Fair. He sees that they are selling bunny rabbits there, and cages. Each bunny rabbit costs $5. Every cage costs $35. John spends $125 on rabbits and cages. John has some cages at home and is just going to buy enough to make sure that he doesn’t ‘overcrowd’ all his rabbits. How many rabbits, did John buy? Call x the number of rabbits. How many cages did John buy? Call each cage y. What we have here is the equation: We’ll call ‘r’ 1 rabbit; ‘c’ stands for 1 cage. xr + yc = $125. We can put in the amount each costs into the equation, replacing rabbit and cage with dollar amounts. X(5) + y(35) = 125.
There is only one correct answer if John only uses the cages he buys at the Fair to house all of the rabbits he buys at the Fair without crowding more than 3 into a cage. However, because John has cages at home, there are 3 answers to this problem. The possible number of cages he can buy is the set Y. Y = {1,2,3} The number of rabbits he buys is interdependent with the number of cages that he buys. When y = 1, then x = 18 bunnies.
When y = 2, then x = 11. When y = 3, then x = 4. The set of correct solutions for x are all put together into a set called X. X = {18, 11, 4}.
The limitations put onto the situation are what make it possible to derive a very manageable number of multiple solutions to this problem. When learning algebraic theory esp. as a child, this can be strange. In this sense it is easier to learn algebra as an adult because you will have had so many more experiences when it has come up and you have used it…not just money, but algebraic real life problems. The two main methods of solving problems that have more than one variable- in this case x & y, are to use substitution. In substitution you just try something real: in this case, you just ask yourself what would happen if John bought one cage. This causes you to get a real answer for y. When you try x = 4, you will find that it won’t work. This let’s you know that you have ‘gone too far’ and that this number is outside of the solution set for x in this problem.
The obvious connection between x & y in each solution to this equation is called a function.
Functions: there is another kind of algebraic equation called functions. These simply express the way things relate with one another. A perfect example of this that can be described in a simple linear formula is laying a brick wall with staggered bricks for solidity. How high is the wall? This relates to how thick each brick is and how many layers there are. Say there are 3 inches per brick. Now you can answer the question: How many bricks are there where the wall is 1 foot high? Function formulae are often expressed in terms f(x) = y. In this example, the rule is that x is the number of bricks, and y is the height of the pile. So the question can be put as: How many bricks make the wall so high? Whenever y=1foot, then we can derive x by our knowledge that each brick is 3 inches. We need the following facts to put this together: 1foot = 12 inches. 1brick =3inches. So, when we divide 12 by 3 then we will get the answer, which is 4. In this case, because the problem is a simple one, you can see that it is very similar to the rabbit and cage problem.
There are names for equations based upon what they do, but also on how they appear.
5x + 35y = 125 is a linear equation having 3 solutions. This is a slight variation on the way the rabbit & cage math problem was written above. Whenever you see the f(x) = y form; that is going to be an equation called a function. Many functions are complex, but others are rather straightforward.
