I received this problem from a friend, who was having trouble while helping her nephew with it. It turned out to be quite a doozy, so I'm presenting it as today's Math Journey to show how the process we used last time works even with a gnarly, complicated problem.
Solve using the Addition Method:
3x – 3y + 4z = – 15
3x + y – 3z = – 8
23x – y – 4z = 0
As we discussed last month, the basic idea behind solving a system of equations is to use one equation to solve another for a specific variable, and to do that enough times that you can eventually rewrite one of those equations with only one variable in it, and solve from there. The way I learned to do this is the “substitution” method, where you solve one equation for one variable, plug the expression in for that variable in a second equation, et cetera until you're down to one variable. The addition method works a little differently, but it's the same basic goal: eliminate enough of the variables...
Settle in, folks, today's a long one.
The Function Machine, we learned that functions can be depicted as curves graphed on a coordinate plane. In
What Does the Function Look Like?, we learned how to tell the general shape of a function's graph based on characteristics of its equation, and vice versa. Today, we'll be focusing on linear equations (meaning any equation that graphs into a straight line).
The defining characteristic of a linear equation is that the highest power of x in the equation is x to the first. This denotes that for every y value, there is exactly one corresponding x value. Of course, there is always exactly one corresponding y value for every x, but this is one of those “square is a rectangle; rectangle is not necessarily a square” moments. We know there's exactly one y for every x because we choose our x's independently and the y's are dependent on them. There can't be more than one y for any given x; you've only got one output slot...