## Simultaneous Equations in the 1800s

Posted by mark schwartz on May 15, 2016

__Introduction__

In today’s Algebra texts, the methods for solving pairs of simultaneous equations are pretty much the same, and they are typically presented in the same order. First is by graphing, next by substitution and then by addition. These three methods seem to be the core ways of solving pairs of equations, and the farther one goes in Algebra, one is presented with matrix/determinants and other ways of handling pairs of simultaneous equations. But there were other methods in the 1800s.

__Story__

When I sampled my collection of old texts I found a method in three books -1864,1866 and 1874 – that was popular but seems to have dropped out of favor and was gone (as far as I can tell) by 1911. These old texts did include substitution and addition but each of them first introduced ** elimination by comparison**, abbreviated to

**Although I’m not proposing that it be reintroduced into today’s curriculum, I find it just as valid as today’s methods.**

*elimination.*Basically, without a direct citation from any of the 1800s texts, the method works like this; ** solve both of the equations for one of the variables and set those two solutions equal to each other.** Perhaps it’s a variation on the substitution method, yet the old texts presented substitution and elimination.

Here’s the example from Ray’s 1866 *Primary Elements of Algebra* (pg. 116) that he used to illustrate the method, not showing every step of the solution.

Equation 1 x + 2y = 17

Equation 2 2x + 3y = 28

From equation 1, x = 17 – 2y and from equation 2, x = (28 – 3y)/2

Therefore 17 – 2y = (28 – 3y)/2

Then y = 6 and x = 5.

It does seem to be a little less efficient than substitution and perhaps that’s why it fell out of favor. Of course, a lot of the “efficiency” can depend on the coefficients and the constants in each equation, but in the long run, solving one equation (substitution) rather than two (elimination) seems more efficient and efficiency seems to be the essence of rules, formulae and algorithms.

There was another method – also no longer presented today – in Greenleaf’s 1864 *New Higher Algebra* (pg. 106) and it was called ** undetermined multiplier.** Here’s the example he used to illustrate the method, not showing every step of the solution. He also has some very interesting commentary after illustrating the method.

Given

equation 1 4x + 2y = 20

equation 2 6x + 4y = 32

Multiply equation 1 by m, giving equation 3, 4mx + 2my = 20m

Subtract equation 2 from equation 3 and factoring with reference to x and y gives

equation 4 (4m – 6)x + (2m – 4)y = 20m – 32

assume 2m – 4 = 0, the second term of equation 4 disappears and

(4m – 6)x = 20m – 32 or x = (20m – 32)/ (4m – 6)

whence since 2m – 4 = 0 , m = 2 and x = 4

Substituting the value of x in equation 1, y = 2

Mr. Greenleaf then continues with this interesting commentary: “in equation 4, x might have been eliminated by assuming 4m – 6 = 0; but the value of m would have been 3/2. Now if we multiply the first of the given equations by 2, the first value of m, the coefficient of y in it will become the same as in the second equation; and if we divide the second of the given equations by 3/2, the second value of m, the coefficient of x in it will become the same as the first. *Hence, this may be regarded as a generalization of the method by addition“ (*my italics).

Be careful here; he’s not saying do both at the same time; it’s one or the other which brings it to being able to use the addition method. Not only has this method disappeared but also – obviously – the idea of a generalization of the addition method has disappeared with it.

I’m not proposing that either of these 1800 methods be revived but the comparison of these two methods with the substitution and addition methods (which were also presented in the 1800s) in my thinking shows that the mathematics domain whittled all the methods down to the two that are seemingly more efficient. I’d also like to note that at that time, none of the 1800 texts I’ve looked at have a graphing solution. I have no idea when graphing came into the picture (pun intended).

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