Algebra 1
The Basics
by Frederick Hoehn, copyright 2014, all rights reserved.
Chapter 1
I previously wrote a book, "Boolean Algebra." Boolean Algebra deals with logic functions. The "and" function, the "or" function and their inversions.
But this is your normal Algebra, as taught in Middle School. Or, if you missed it in middle school, they also teach it in College, where it might be called "Algebra 101."
Algebra was one of my favorite subjects in school, and I did well in Algebra. One thing that helped me to do well was a statement by my Dad, when I was a boy. He said, "In Algebra, I was number one."
It helped me that he said that. Because he had been number one, and because I was his son, I expected to do well at Algebra, and I did.
If you're a Christian, say this out loud with your mouth, "I can do all things through Christ who strengthens me, and I'll do well at learning Algebra, and at all my endeavors." (Phil 4:13)
Your expectation that you'll do well will help to cause that you'll do well.
People are built that way.
A saying attributed to Henry Ford is, "Whether you believe that you can do it, or whether you believe that you can't do it, either way, you're right."
This ordinary kind of Algebra uses a letter, such as capital X, to represent an unknown quantity. Information about the problem is put in the form of an Algebraic equation, and then mathematical operations are performed on the equation, solving for X.
An equation is a mathematical expression that includes an equals sign (=). The part of the equation to the left of the equals sign is equal to the part of the expression to the right of the equals sign.
Here is an example: A man drives his car at a constant speed of 60 miles per hour on the Interstate Highway starting at the city of origin to a destination city that is exactly 90 miles from the city of origin. For how long will he drive?
Now, you could probably solve that problem in your head, and so could I, but let's use Algebra to solve the problem.
We have a basic formula from Physics that "Distance = Rate x Time." Spoken out loud, "Distance equals the rate times the time."
If you multiply the speed of travel by the time expended, you get the distance traveled.
Now let's use that formula to solve the above problem. It is a given that the rate is 60 miles per hour. We know that the distance is 90 miles. Now let's plug those values into the equation.
Here is our equation for the above problem: 90 mi. = 60 mi/hr x X
The unknown here is X, that is being used in the equation to represent the time.
To find the time, we solve for X.
There are four basic axioms in Algebra that say, 1) "If equals are multiplied by equals, the results are equal." 2) "If equals are divided by equals, the results are equal." 3) "If equals are added to equals, the results are equal." 4) "If equals are subtracted from equals, the results are equal."
But remember this, in mathematics, division by zero is not allowed. So if you want to divide both sides of an equation by (a - b), and if a happens to be equal to b, then you're dividing by zero, and that's a no-no.
The equation says 90 = 60 x X
Since we can divide both sides of an equation by equals, lets divide both sides of this equation by 60.
That gives us a new equation, 90/60 = X.
But it's customary to put the X on the left side, so,
X = 90/60.
When you divide 90 miles by 60 miles per hour, you get 1.5 hours. One and one half hours. That's how long the driver will drive at the rate of 60 miles per hour to go 90 miles.
And though you could have solved this problem in your head, there will be other problems not so easy to solve in the head, where a knowledge of Algebra will come in very handy.
Chapter 2
Now there are certain rules regarding order of operation for mathematical expressions. Consider the following expression--
2 x 7 + 5
Does this expression mean we should double seven plus five, which is twelve, and get twenty four?
Or does it mean we should multiply two times seven and get fourteen, and then add five, obtaining nineteen?
We get two different answers, depending on the order of operations.
So here are some rules about order of operations. These rules are conventions. These are the conventional way of doing things in mathematics so that mathematicians (or, Algebra students) will get the same answers all over the world.
Step 1. Do the operations that are enclosed in parentheses.
Step 2. Calculate the value of any exponential expressions. If you have an X squared in your expression, that is an exponential expression, and means that X is raised to the second power. You'll see the X with a small 2 a little above and to the right of the X. In other words, X is multiplied by itself. X raised to the third power would be X multiplied by X multiplied by X, and is also called X cubed. 2 squared is 4. 2 cubed is 8.
Step 3. Multiply and/or divide from left to right.
Step 4. Do the additions and subtractions.
When we use X to represent a number, X is called a variable because it can take on many different values.
Here is a mathematical expression:
abc + bcd + bdf + aad
The above expression is said to have four terms. The terms are the short expressions connected together by the plus signs (or, sometimes minus signs).
In Algebra, we use the four axioms given above to do mathematical manipulations to solve for the unknown quantity, often represented by the capital letter X.
Now here is another thing that can be done called transposition.
Suppose we have the expression,
X + 5 = 0
Transposition allows us to move the 5 to the other side of the equals sign, reversing its sign from plus to minus.
We then have
X = -5
Transposition is derived from the four axioms, and makes life a little easier when solving problems.
Now you can add up things that are alike. You can add your numbers. You can add X's to X's and Y's to Y's, but you can't add X's to numbers because we don't know what X is until we've solved the problem. You can have an expression, X + 5, but we can't actually do the addition while we don't know what X is.
Here is another Algebra problem:
If a man has $10.00 to spend on potatoes, how many three pound sacks of potatoes can he buy if each sack costs $2.79?
Of course, you could get the answer without Algebra, but let's use Algebra to solve the problem.
First we make an equation.
X = $10.00 / $2.79
Doing the division, we get 3.5842.
But probably the supermarket won't let you buy a fraction of a bag of potatoes, so the answer would be 3.
A very simple problem.
Let's find some more difficult problems.
If gasoline is $3.59 a gallon in this city, but the handicapped driver has to pay an extra $1.50 to have the gas station attendant pump the gas, how many gallons of gas can he purchase for $15.00?
X = ($15 - $1.50) / $3.59
Remember we must first do the operation inside the parentheses. So, 13.50/3.59 = 3.76 gallons of gas.
The square root of a number is the number which when squared yields the original number. Thus, the square root of 25 is 5, and the square root of 36 is 6.
The easiest way to find square root is with a pocket calculator. I happen to have a Casio calculator that I paid around $12 for at Walmart. It also does Trigonometry functions, beside square root and ordinary math calculations.
To find square root, enter the number you're finding the square root of, and you'll see that number in the display. Then press the square root key.
The square root key is kind of like a V with a horizontal line extending to the right, off the upper right corner of the V.
After pressing the square root key, the display will show the square root.
When you get the square root of a number the first time, then take that number and multiply it by itself (squaring it) and see if it's very close to the number you were taking the square root of.
So, this proves that you were doing square root correctly with the calculator.
Square root can also be found using logarithms, or by using a slide rule, or by using mathematical tables, such as the CRC Standard Math Tables.
If you have a calculator that doesn't do square root. Just take a guess at what the square root might be, and then square that number. If it comes out too high, then decrease you guess, and square that.
After enough guesses, you'll come out pretty close to the square root. This method is called successive approximation.
But I figure you'll be able to purchase a calculator that does square root, and that's the easiest way to find square root, most of the time.
Problem: The meat buyer for a local hamburger stand has a budget of $100 per day for Ground Chuck. The local butcher shop has a special on Ground Chuck. The first 10 pounds are $4.25 per pound. The next 10 pounds are $3.95 per pound. Anything above 20 pounds is $3.70 per pound.
How many pounds of Ground Chuck shall the buyer order to spend his $100.00?
Solution: Make an equation. Let X be the number of pounds to order.
100 = (X-20lbs)(3.70) + (10lbs)(4.25) + (10lbs)(3.95)
To the right of the equals sign is a three term expression. In each term, the two sets of parentheses are understood to be multiplied together.
Now, we solve for X. Those two right terms multiply out to 42.50 and 39.50, totaling 82.00.
The first term with the X multiplies out to 3.7X
- 74.00
So, we have 100 = 3.7X + 8.00
Using our axioms, 3.7X = 92.00
Now, we divide both sides by 3.7
So, X = 92.00 / 3.7, which is 24.86
The buyer orders 24.86 pounds of Ground Round, and pays $99.98
Problem: A Painter needs to stock up on his Ivory White paint. He has $200 to spend on the paint. The hardware store has Ivory White on special for $8.99 a gallon for the first 10 gal, and $7.99/gal for the next 10 gal, and anything above that, $6.99/gal.
How many gallons should the Painter buy?
Solution: Make an equation.
200 = (X-20gal)(6.99) + (10gal)(7.99) +
(10gal)(8.99)
Those two right terms work out to 169.8
The X term multiplies out to 6.99X + 139.8
Using our axioms, we have
6.99X = 170
X = 170 / 6.99 = 24 gallons.
The Painter buys 24 gallons of Ivory White, and pays $197.76