Equations

Equations

Algebra is the science of equations. That is, it tells us how to work statements that tells us when something is equal to something else. This is tremendously convenient, especially when you are trying to figure out an unknown quantity. With algebra, you can use a known quantity to get to the unknown quantity.

There are four basic equations in algebra, corresponding to the four operations that you do in arithmetic. Those operations are: addition, subtraction, multiplication, and division.  The four basic equations involve addition, subtraction, multiplication, and division.

They are:

A+B=C

A-B=C

A(B)=C

A/B=C

The way to solve them is to do the opposite of the indicated operation. That is, in an addition equation, you subtract, in a subtraction equation, you add, in a multiplication equation, you divide, and in a division equation you multiply.

This is called the Law of Inverse Operations.

It’s a simple concept to grasp. If you have $10 and I gave you $10 more, you would have $20.  If I were to take that money back, you’d be back to $10. So, if I wanted to figure out how much money you had to begin with, I’d set up the following equation:

A+$10 =$20.

To figure out what A, the original amount, was, I’d subtract $10 from both sides of the equation and I would be left with $10.

Similarly, if someone stole $10 from you, leaving you with $10, I could restore your wealth by adding $10 back.

Multiplication works the same way.

Suppose the eight of us were to invest $8 each. A math wiz would multiply each of our holdings by $8 and come up with a total investment of $64.

If we wanted to know the original amount, we could divide the $64 by 8 and come up with it.

Similarly, if we were to divide $64 by $8 and get $8, but wanted to reverse that, we would multiply by $8.

To solve an equation, we reverse the operation that is indicated by the equation.

To solve an equation, you do to the one side the same as to the other.

You will not be able to solve anything if you only work one side. If X +4 =16, you will never, ever learn what X was by simply subtracting the 4 from the left side. You have to do it to the right to get the answer.

Equations are a form of balance. If you and your friend are on a teeter-totter and President Trump sits on your friend’s side, you will go flying. Unless you have Chris Christy, former governor of New Jersey, sit on your side. In learning algebra, always keep the teeter-tooter in mind.

Now that we got that straight, what do we do when an equation requires more than two operations?

We do them in steps. Do the steps have any order?  Yes, they do. Remember PEMDAS? (Please Excuse My Dear Aunt Sally?) In other words, we do parentheses, exponents, multiplication/division, then addition/subtraction when doing multiple operations.

In solving an equation, we reverse that. Why? Because we want to get back to the original point. So in the following two-step equation:

2X +4 =16

We first reverse the four. To reverse addition, we do subtraction. So we subtract four from both sides.

We get:

2X =12.

Now we reverse the multiplication. We multiplied by 2, so we divide both sides by 2.

We get:

X=6.

What could be simpler?  Well, actually, one-step equations which we discussed at the beginning. Can it get more complicated? Oh, yes.

We’ll deal with that later.

Dr. Fred Young

http://www.agapequalitytutorial.com

 

 

FUNCTIONS

FUNCTIONS. WHAT THEY ARE AND WHY THEY MATTER.

Functions are a crucial concept in mathematics. You get them in algebra, in geometry, in trigonometry. You hardly ever get them at parties, but that would depend on the nature of the party, I would guess. Today, I want to explore some of the properties of a function.

A function consists of a correspondence between a first and second set. Each element in the first set corresponds to exactly one element in the second set.

Take the following:

A 1
B 2
C 3
D 4
E 5

 

This table denotes a function. For every item in the first column, there is a corresponding item in the second.

The first set is called the domain, the second, the range.

The domain is also the x-axis of a set of Cartesian Coordinates; the range is the y-axis.  This means that you can make a graph of a function!

Image result for graphical depiction of a function

These are graphical depictions of functions.

Now, suppose you were to draw a vertical line through one of the lines already on the chart. It would be a function. As the line represents a point on the x-axis and goes through exactly one point on the y-axis. This is called the Vertical Line Test of functions.

The above graph shows the use of functions. If we had two sets of data, we could see correlations.

Say the x-axis represented increased gun violence and the y-axis gun ownership. The first would show a direct correlation between gun violence and gun ownership; the second an inverse correlation.

CNX_Precalc_Figure_01_01_0122.jpg

Notice the vertical lines. The first defines a function, the second and third do not.

coulson_nat_ed_trends_chart_sept_2014.gif

This graph isn’t a purely functional graph. The y-axis is a bit crooked. But it would show a one-on-one correlation between the x-axis and the y-axis. (It also shows that as spending on education goes up, absolutely nothing happens to students’ skills, but you already knew that.)

And here’s a graph that is called a “best fit” graph, as you can’t draw a straight line.

 

mass shootings.jpgHowever, if you did draw a straight line, which is possible on the incident line without too much distortion, you would have a function. Mass shootings were 25 a year in 1980 and about 23 in 2010.

As should be clear by now, functions can be expressed by linear equations.  But instead of:

Y =2x+4

You say instead:

F(x)=2x +4.

We’ll continue this discussion in a future blog.

Dr. Fred Young

http://www.agapequalitytutorial.com

 

 

COIN PROBLEMS

COIN PROBLEMS

 

Word Problems.  Everyone’s favorite. From time to time, I will blog on how to solve them and other algebra problems. Today’s blog will be on coin problems. You know, the ones where you wonder why don’t the people simply look in their pockets and count? But they hone your algebra skills and allow you to understand principles involved in systems of equations and other mathematical procedures.

Here’s a typical one:

Jack and Diane have 28 coins that are nickels and dimes. If the value of the coins is $1.95, how many of each type do they have?

As with many math problems, once you simplify, the answer is a lot easier to come by. We have two variables. Ah, if we only had one.  But alas, we have two.

The first step is to get rid of that pesky fact and define one in terms of the other. This way, we only have one. Don’t worry. We will later bring back the other variable.

The variables will stand for our two coins—nickels and dimes. So our variables are N and D. (Pretty clever, right? I could have picked X and Y, but N and D are so much more representative of nickels and dimes, I think.)

The problem tells us that the number of nickels and dimes adds to 28. So our first equation:

N +D = 28.

The number of nickels, N, and the number of dimes, D equals 28. With me so far?  Good.

But we need another equation to solve our problem. The solution is to define the variables and the result in the equation (in this case, the $1.95) by elements they all have in common.  They are all composed of pennies. Hey, that leads us to:

5N +10D =198.

As there are 5 pennies in each nickel, ten in each dime, and 198 altogether. Beautiful. Now, all we have to do is to define one variable in terms or the other and we are all set.

Let’s define nickels in terms of dimes. We could do it the other way, but I like nickels. But maybe you like dimes. Then, you can do it for dimes.

N + D =28.

We solve for N.

N + D-D=28-D

We add the -D to D on the left, and what happens when you add the additive inverse to a number? You cancel the number. As we are in an equation and have to do to the other side what we do to the side we are working, we get the result 28-D.

Think of an equation as like a teeter-totter. You start with your friend who equals your weight. Then Donald Trump sets on your friend’s side. You go flying. Unless, of course, Chris Christy sits on your side. You are back in balance. All’s well, the world, or at least the equation is in balance.

So, N =28-D.

We substitute that for N in our equation:

5(28-D) +10D =195

We now multiply both terms of the parenthetical expression by 5 and the result is:

140-5D+10D =195

We combine like terms.

140 +5D =195

We now have your classic two-step equation. You solve it in, well, two steps.

Step 1

140-140 +5D =195-140

There’s that teeter-totter again. Maybe a less controversial example would be Nero Wolf and Cannon, two fine but large detectives. In any case, however you imagine it, you get:

5D =55

Step 2

Now, to cancel a multiplication, you do a division. To cancel five times D, you divide by 5. Again, you do it to both sides and you get:

D =11.

And Since N+D =28, and you can now substitute 11 for D, you get:

N+11 =28

You subtract the 11 from both sides, and you get 17

Next time, we might try something really hard, like how to balance your checkbook.

Dr. Fred Young

http://www.agapequalitytutorial.com