Framing equations from the scales

Go back to the weighing-scale pictures. You solved them by balancing and removing. Now frame EQUATIONS for them, using letter-numbers to denote the unknown weights, and see that the two routes agree.

Take the fried-egg scale. Let the weight of one fried egg be e. Each slice of bread is 2, so one side is 2 + 2 + 2 = 6, and the other side is e + e. Since they are equal, 6 = e + e, or 2e = 6.

Take a second scale. Here one shape is known to be 4, and we denote the weight of the unknown shape as y. There is 16 on one side, and 4 + 2y on the other. So the equation is 4 + 2y = 16.

Notice what the framing does. The picture says 'these two plates balance'. The equation says the same thing in symbols — and once it is in symbols, the rules of arithmetic can act on it.

Now solve the equations you framed and CHECK whether you get the same value for the unknown weight that you got before. If the picture gave e = 3, the equation 2e = 6 had better give e = 3 too. It does, and that agreement is the point.

Do this for every scale in the section, then go further: frame 5 equations of your own, and find methods to solve them. Some will be easy to see; others will not, and those are the ones that make the next section necessary.

From here on the weighing scale can be put away. Everything it taught — that both sides stay equal, and that you may do the same thing to both — carries over to equations written in symbols.