Guess and Check Method for Simultaneous Equations: Shown to Me by Kai Alton.

Here is a typical problem with two unknowns:

The gym has 2 kg and 5 kg disks for weight lifting. There are 14 disks in all.

The total weight of the 2 kg disks is the same as the total weight of the 5 kg disks.

What is the total weight of all the disks?

The first step is to read the problem and turn the English into mathematics. Part of this is: give the unknown quantities one-letter names that are meaningful. Also: identify what they are asking for. In this problem, it is the total weight of all the disks.

The two unknowns are the number of 2 kg disks and the number of  5 kg disks. Let’s call these T for number of  Two kg disks and F for number of Five kg disks.

The first English sentence to turn into mathematics is: “there are 14 disks in all.”

In mathematics, it becomes T + F = 14.

The second English sentence to turn into mathematics is: “the total weight of the 2 kg disks is the same as the total weight of the 5 kg disks.” In algebra, if we write a number next to a letter, it means “multiply.”

So the sentence becomes: 2 T = 5 F.

There are formal mathematical methods for solving the two equations simultaneously, but I would like to show you a method shown to me by a Seward student, Kai Alton. It works best where the total of the two unknowns is given, as in T + F = 14.

First:  subtraction <= undoes => addition (read it forwards or backwards).
Also: division <= undoes => multiplication (read it forwards or backwards).

A summary of the method is:

1. Make a smart guess for one unknown.
2. Solve for the other unknown in terms of your smart guess, using the “undoing” rules.
3. Check to see if the two (now known) unknowns add up to the given total.
4. If not, make another smart guess until the total is right.
5. Use the (now known) unknowns to give them what they are asking for.

Using the “undo” on 2 T = 5 F, divide both sides of the equation by 5, undoing one multiplication.

The result is F = (2/5) T . Both F and T are whole numbers, so a smart guess for T is a whole number divisible by 5, for example, guess T = 5. Then, F = 2 and the total number of weights is 7. The total should be 14, so a better guess is T = 10, which gives F = 4 and a total of 14. Checking,  5 F= 5 x 4 = 20 kg  and 2 T = 2 x 10 = 20 kg (the total weight of the 5 kg weights equals the total weight of the 2 kg weights) and what they asked for, the total weight of all the disks, is 20 + 20 = 40 kg.

The problem is solved by smart guessing, checking, and arithmetic.