Last week, in Fundamental Understanding of Mathematics XLVI we investigated why equals multiplied by equals are equal. In earlier diaries, I wrote about the associative property for addition, the commutative property for addition and multiplication, the distributive property of multiplication over addition, the identity property of addition and multiplication...
substitution of equals, three properties of equality, reflexive, symmetric and transitive, and, of course, that equals added to equals are equal.
These are the nuts and bolts, pulleys and gears, belts and springs and widgets we use to manufacture the machinery of algebra: the symbolic manipulation of equations. With these dozen parts, we can build most of the algebra that is laboriously memorized in modern American math classrooms. Let's take a look at some of it.
Simplifying expressions
Sometimes stuff on one or the other side of an equation is not simple. For example:
2x + 3y - 2(x + 2y) + 17 = 2x + 5
The right side of the equation is pretty simple, but the left side is a mess. Let's take a closer look at the left side.
2x + 3y - 2(x + 2y) + 17
If we replace the unknowns with fruit, it might be easier to see what's happening here. Let x be an apple, and let y be a banana.
So, we have some apples and some bananas, and we subtract two groups of apple and bananas, and there's also just a number added in at the end. We can deal with the two groups by doubling everything in the group, but we keep it in a group because we are going to subtract the whole thing.
Let's give these things some names, to make it easier to write about them. I've already identified the apples and bananas as unknowns. Most text books below college level call these things "variables," but that's a misnomer. They represent fixed values, we just don't know what the value is, yet.
Something that varies, a variable, is something that changes over time, like the weather or your mood. If we find out that our apple represents, say, 14, we don't expect to return to this expression at a later date and discover that the apple represents 12. Whatever number it is, we expect that number to be fixed.
Telling students that these "could be any number" does not mean that the number might change, it simply means that we don't know what the number is. Calling these things "variables" says that maybe they could change, after all, that's what variables do. They change. Using the word "variable" to describe an unknown value sends a confusing mixed message at a time when it pays to be clear and precise.
Now, we also notice that we have groups of one or more (in this case more) apples and bananas. To save some space, instead of
we could write
This is the same as writing two multiplied by apple. We may not know how to multiply by apple, but we can multiply by 2. It is the same as two apples, our earlier diagram. We call that number, 2, the coefficient. Let's redo our diagram with coefficients
Then we have that integer, 17, added on at the end. Most textbooks call this number a "constant" to distinguish it from the apples and bananas. Now, most of them do this because they call the apples and bananas "variables," but we've decided not to do that. We could keep things simple, and just call it "the integer," but it doesn't have to be an integer. It could be a decimal number or a fraction. What does distinguish it from the rest of the expression is that it is completely known. There are no unknown parts to it. So we may as well call it a constant, just to be consistent with other textbooks, just so long as we remember this does not mean that the other numbers, the unknown numbers, are variable.
Then, we have terms. Terms in an equation are small groups that, when we use the commutative property to move things around, must be moved around as if they were a single thing. So,
here are the terms in our expression (2 apple is only shown once, although it appears in the expression twice.)
Finally, we have groups. Groups are collections of terms that we treat as if they were terms, and we show groups by putting a parentheses around the terms.
is our expression's only group. We use the distributive property or the associative property to group or ungroup terms. The associative property lets us make groups pretty much at will, the distributive property has specific rules for creating or removing parentheses.
So, here we are. What would a simple form of this expression look like? Well, fewer terms, for one thing. This expression has five terms, or four terms, one of which is a group. If we can cut it down to three terms, it would be simpler.
From inspection: we start with two apples, then later we subtract two apples. We will end up with no apples. We also start with three bananas, and we will subtract four bananas. We will end up minus one banana. And of course, we will end up with the constant, 17. So, the end result of simplifying this expression will be 17 minus one banana.
Now we have the goal, how do we get there using properties we know about numbers?
Well, this is about symbolic manipulation of equations. Manipulation. We to want to move things around. We can use the commutative property for addition to move things around, once we get rid of that pesky subtraction, because subtraction isn't commutative (4 - 3 is not 3 - 4.)
Fundamental Understanding of Mathematics XVI explained how changing subtraction to adding an opposite works. We'll use that here:
Now we can use the distributive property to get rid of the group 2 apples plus 4 bananas.
Now we have a series of additions, so we can change the order of terms using the commutative property
Now we can group our apple terms with each other, and our banana terms with each other. In addition to doing that, I'm going to change the "adding the opposite" back into subtracting, because I'm looking forward to using the distributive property, which works equally well with subtraction as with addition, and it will get rid of some extra parentheses.
We saw this one coming: here is two apples minus two apples, and three bananas minus four bananas. With a 17 added on the end.
One more step, just to be pedantic: we're going to factor out the unknowns, using the distributive property in reverse [(ab + ac) = a(b+c)]
Now our coefficients are grouped, and are simple arithmetic problems. So we solve those, which gives us zero apples, and minus one banana. The identity property for multiplication tells us we don't really need to write the 1 for minus one banana, we can just write minus banana.
Pretty simple, and what we expected to get when we started. Let's remove the banana by recalling that the banana is the "y" unknown quantity, and bring back the right side of the equation.
-y + 17 = 2x + 5
Much simpler. We still don't know what x and y are, but we didn't set out to solve the equation, we set out to simplify the expression on the left hand side.
Now, let's solve it.
When we deal with solving equations, we can use any property or principle that preserves the main point of an equation: that the two sides are equal. We have a couple of tools in our inventory that directly relate to that: equals added to equals are equal, and equals multiplied by equals are equal.
Our simplified equation is, by definition, equal on both sides. Any changes we make can change the values, but must make sure that, whatever the values become, they remain equal to each other.
So: y = y. Let's add that to both sides
y - y + 17 = y + 2x + 5
Why did I pick y = y? Well, I want to get some arithmetic to do (similar to simplifying expressions, if I can reduce the complexity of the equation by adding, subtracting, multiplying or dividing known numbers, I'll do it.) So I want all the unknown numbers on one side of the equal sign, and all the known numbers on the other. Then I can do some arithmetic with the known numbers and find out what the unknowns are equal to. I chose to add y because I had a - y on one side, and y - y is zero, so I can get rid of the subtraction.
17 = y + 2x + 5
My goal is to get all the known numbers on the same side, to do that I'll add (-5) to both sides (by our principle that adding opposites is the same as subtracting, I can simply subtract five from both sides.)
17 - 5 = y + 2x + 5 - 5
I've got some simple arithmetic on both sides of the equation, now. 17 - 5 = 12 on the left, and 5 - 5 = zero on the right. Notice how I'm chosing numbers to get that zero on one side or the other.
12 = y + 2x
Here's the solution. But instead of finding a single value for x and for y, we have found a relationship between the two numbers. If x is one, for example, y is ten, because 10 plus 2 times 1 is 12.
We can make that arithmetic clearer, by doing one more bit of manipulation: moving x and y to opposite sides of the equal sign, by subtracting either from both sides. I'll subtract 2x.
12 - 2x = y + 2x - 2x
12 - 2x = y
This is one way to show our solution. We can also write a table of values, by setting x equal to a series of simple numbers, and calculating what y would be in each case.
We can also take those x and y values, and plot them on a graph.
Have fun in the comments.