Systems of Equations

In my word problem slogging I accidentally jumped ahead and worked on this without even knowing it.

A system of equations means they share the same x and y values. The equations themselves are not equal to each other. There are three ways to solve them.

Graphing: Slap the equations on a coordinate plane to find your answer.

Pros: Visual confirmation of (x,y) values.

Cons: Good luck eyeballing the difference between 4.33 and 4.4

Naturally algebra books rig it so any problems you're instructed to solve by graphing will have nice round non-decimaled numbers.


Substitution: This is my personal favorite. Solve one equation for a variable (usually whichever's easiest to isolate), plug it in to the other equation, get a numerical value for a variable, then use that value to solve for the remaining variable. I made that more confusing that is really is, didn't I? Let me demonstrate.

Solve system x - 4y = 11

                      3x + 7y = -5

Since x has no coefficient in the first equation, let's solve it for x.

x - 4y = 11

x - 4y + 4y = 11 + 4y

x = 11 + 4y

Now take x = 11 + 4y and use in the second equation. No, you can't use it in the first equation. You'll end up with 11 = 11. Even I know you can't do that. How did I know that? Never mind...

3(11 + 4y) + 7y = -5

33 + 12y + 7y = -5

33 + 19y = -5

19y = -38

y = -2

Whether you want to solve for x in the first or second equation is up to you. The first one looks easier to me.

x = 11 + 4y (Why go here instead of standard form? Do you really want to move the ys around again?)

x = 11 + 4(-2)

x = 11 + -8

x = 3

The coordinate pair for this system is (3,-2)

Elimination Method: I get it, but I don't use it unless instructed. Multiply one or both equations by a real number, set it up like an addition/subtraction problem, and eliminate a variable.

Solve system 2x - y = 13 

                     6x + 4y = 4

Greatest common factor time! You can do either 4(2x - y = 13) or -3(6x + 4y = 4). Why -3? If you go that route your goal should be to eliminate the 6x, for which you'll need it's opposite. For some odd reason I went with -3 even though I walk the path of least resistance.

 6x + 4y =    4

-6x + 3y = -39

 0x + 7y = -35

y = -5

Solve for x

2x - (-5) = 13

2x = 8

x = 4

The coordinate pair for this system is (4,-5)

Where Did My X and Y Go, and Should I Care?

Sometimes variables disappear when solving systems of equations. Check your math FIRST before classifying it under one of these two exceptions.

1. If the ending statement is FALSE (7 = 5), the system has no solutions and their lines are parallel on a graph. If you're not staring down a classroom clock, graphing the equations is a good way to confirm it. If you wind up with intersecting lines, you messed up somewhere. Please don't ask me to figure out where.

2. If the ending statement is TRUE (10 = 10), the system has multiple solutions and is inconsistent. This means their graphs overlap and the equations are part of a dependent system. They'll never break up with each other because the crazy they know seems safer than unknown crazy.

Systems of Inequalities

Both Everything and Idiot's Guide solve systems of inequalities by graphing them, shading overlapping regions, and picking test points within the answer region to confirm. Everything tells you how to plot it on a TI-81. I don't have a graphing calculator, nor do I ever plan on owning one. If I recall correctly, learning how to program those suckers is almost as frustrating as learning the math. Sorry, can't be of much help here.

So...I have an admission to make. My blogging hasn't slowed because I've stalled in the material. Quite the contrary. My blogging's stalled because I got my daughter Lego Star Wars: The Complete Saga for her birthday. And by got it for her birthday, I mean bought it for us to play together under the pretext of a birthday gift. And by play together, I mean she watches me play because her motor skills aren't good enough to handle the controls yet.

Okay, fine. I like Lego Star Wars. I've only cared about two video games in my life. This is one of them. (The other is the original 8-bit Tetris, but only if I have to blow on the cartridge.)

Monday and Tuesday I waded through matrices and Cramer's sadistic Rule. Today I got through adding, subtracting, multiplying, and dividing (both long and synthetic) polynomials. The matrix post will take some time. It requires a lot of drawing in Photoshop and formatting. All About Polynomials should be up tomorrow.