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Question

Dear Mitch,

The other day someone asked a question about doing "Systems of equations". The person was a teacher and so you explained some way to make up problems of systems of equations. But I'm not a teacher, I'm a student and I did so bad in math (but just some of the stuff we did) that they gave me a bunch of review stuff to do over the summer and told me they'd test me in September to see if I could go on to the next grade or if I'd have to repeat the whole year like a total loser.

Anyway, one thing I could use help in is systems of equations. I know there's a lot to it, because the class spent forever on it, but is there anything you could tell me like one of your tricks that would help met get the idea?

Thanks,

Mitch Y. (SAME FIRST NAME AS YOU!, though I usually go by "Mitchell")

Answer

Dear Mitch/Mitchell,

Either way, nice name!

You are correct in noting that there is a lot to the topic of systems of equations, but the basic ideas behind them are actually neither complicated nor comprised of so many different concepts. It's just that the style of questions vary, and there are many different ways to solve them.

Here are a few of the basics that I think should help the topic make sense to you:.

1:

All that a "system of equations" is, is a set of equations (two or more), and almost always with more than one variable. (It's a matter of debate whether a set of equations with only one variable between them constitutes a real "system of equations", but when it comes to semantics or technicalities that have no bearing on how you do the work, who cares what people call it? 

 

2:

The first basic type is the two equation set with two variables

Like this: a + b = 43

15b - a = 5

 

3:

The basic idea is that you want to find a way to somehow combine the two equations or combine parts of the equations so that you end up with one equation that has only one variable instead of two.

NOTE: This part, part 3, is what feels strange to a lot of students. The reason it feels strange and usually looks strange is that every since you started doing math at the middle school level you have heard the word simplify, and you probably heard it many hundreds if not thousands of times each year – YET, this step, step three makes everyone new to systems of equations feel they're doing something wrong, and moving in the wrong direction, because instead of ending up with something that LOOKS simpler, the way the work usually progresses in a textbook, you end up with something far uglier and more complicated-looking than you had only one step before, BUT HERE'S THE THING: this is one of those times when mathematically simpler is not necessarily the same thing as LOOKING simpler. Your work after step 3 will look much more complicated, but it IS simpler in one important way: Even though your new equation will be longer and weirder-looking, it will have only one variable instead of two. And that is the critical difference, and the critical improvement.

 

4.

The most common way to accomplish the task of step three is to pick one of the two equations and rearrange it so that you have one letter by itself on one side of the equal sign, and every other part of the equation on the other side. Such as:

                                     X = 2Y+ 65.

 

5.

This means you now have a new name for x, which would be 2y + 65.

And since 2y + 65 equals x and is a new name for x, you may now substitute x's new name anywhere in the world where you happen to see an x. It's like this: if I changed the name of something in the real world, such as stapler, and found a way to get the item to be called an plubdubbler, then wherever in the world the word stapler appeared or wherever there was a staler, I could replace them with plubdubblers and since they were exactly equal the same, no one would find the world's balance to be off or even effected. People could still staple papers together with their plubdubblers, which would look and work exactly the same, and all would be as well as before.

So, again, you take your new name for the letter you've defined as a mishmash of the other letter and substitute it everywhere the other letter previously appeared.

BUT NOTE (AND THIS IS KEY): when substituting your newly named version of the variable, do not stick it into the equation you used to rename one of the two letter you renamed. Otherwise you could end up with an endless loop in which nothing new is learned.

So you go to the other equation (the one you did not use to rename a variable) and plug in the "new name" wherever you see the variable that you've renamed.

Example:

If you have a + b = 22,

You can change it to a = 22 – b

Then wherever you see an "a" in the other equation, you plug in (22-b).

 

6.

Next, you take your new (and longer) equation with only one variable, and simplify it to solve for that variable

 

7.

Next, you take the number you get for that variable and plug it into whichever of the two equations looks to be easier and then solve for the other variable (note, you will once again be using an equation that has only one variable, the one yet to be solved.)

SO:

Now that you've got the basic idea of the basic way to do them, here is a "trick":

You can simply line up the two equations vertically, i.e., one under the other with the variables in the same vertical columns, and subtract one entire equation from the other to get rid of one of the variables, leaving you with that magic one equation with one variable. Solve it and then plug that answer into either of the two original equations to find out what the other variable equals.

NOTE: Often, before just subtracting one equation from another, you have to multiply one of the equations by some number so that subtracting them will in fact get rid of whatever is annoying you.

EXAMPLE:

B + 2J = 94

3B + J = 82

If you multiply the top one by 3 then you will have a 3B in the top one and a 3B in the bottom one so subtracting one equation from the other will get rid of your B's

Like this:

3(B + 2J) = 3(94)

3B + 6 J = 282

Now (if possible) try to line them up so that the bigger numbers are on the top:

  3B + 6J = 282

- 3B + J = 82 (Note: This one didn't change; it is the one we were given.)

____________

0 + 5J = 200

5J = 200

J= 40

Plug that 40 into one of the original equations:

            B + 2J = 94

B + 2 (40) = 94

B + 80 = 94

B = 14

Then you have to check your answers by trying your J and your B in both original equations to make sure everything works out. I'll leave that up to you. But to be honest, I'm not too worried about it!

Hope this helps,

Mitch