Introduction To Algebra
This lesson consists of providing you with a Self-Tutorial on what is algebra, what are variables, constants, coefficients, terms, and expressions. I explain the use of proper notation, how to combine like terms, find the negation of an algebraic expression, how to evaluate an expression (by hand and by using your calculator), and finally, there is a VERY detailed section on how to translate English phrases into algebraic expressions.
Step 1: Introduction
Hello. My name is Luis-Anthony Ast, and I'm the video math tutor. Welcome to Introduction to algebra. Before we get started, please take the time to go over the basic math videos that are on my website. They're all free, so do take your time. To do well in algebra, it's very good to have a very firm understanding of arithmetic. Also, I would like you to take the time to print up these notes, and to go over them before you actually go through this video. That said, lets get started!
Step 2: What is Algebra?
Algebra is the area of mathematics that generalizes the concepts and rules of arithmatic using symbols to represent numbers. The symbols used are called variables and constants. Here are some examples of symbols that are used in algebra. They can be letters from the English alphabet , like the x, the A, the e, or the z. Or they could be letters from other alphabets. theta and pi are typical examples.
Step 3: Symbol Representation
A variable is a symbol that can represent a value that is not known, or a symbol that can take on different values from a given set of values. Traditionally, the symbols most often used to represent variables are the lower case variables at the end of the alphabet. So, the s,t,u,v, w, x,y,z . The letter x, by far, is the most commonly used variable. Contrary to what most students believe, it isn't evil. It's just a variable. Sorry. Any letter can be used to represent a variable. It really just depends on how its being used in a problem.
Step 4: Value
A constant is a number or symbol that represents a specific value. The value of piis the most well known mathematical constant. It really just represents the number of diameters that fit in the circumference of a circle. Letters can also be used to represent constants. Traditionally, they are either lower case or upper case letters at the beginning of the alphabet.
Step 5: Addition
A constant being added to a variable is traditionally written to the right of the variable. Here are some examples. The x,the y, and the w are the variables. The 3, b, and pi represent the constants.
Step 6: Multiplication
A constant being multiplied by a variable is usually written to the left of the variable.Here are some examples. The two, the five, and the A are the constants. Z, w, and x are the variables.
Step 7: Times Sign
When multiplying constants times variables, dont use the times sign. Why? Well, it can be confused with the x variable. Instead, I recommend you use parentheses like this, or a raised dot. Like this. Better still, just put the constant next to the variable - like this. This represents implied multiplication. To further illustrate our point, lets take a look at this. We can have , things that are OK, are implied multiplication, the raised dot, the parentheses. Try to avoid things that look like this. Again the times sign, with the x's. The constant should not be to the right it should be to the lefttraditionally multiplying . Parentheses just around the constant, it just doesn't look good. Traditionally, its just not done. Since order matters for subtractions and divisions, variables may appear in any place. It just depends on how they are being used in a problem.
Step 8: Negative Sign
A coefficient is a constant that is being multiplied to the left of a variable or series of variables. Now lets identify some coefficients. The first one, the coefficient would be just 2. This, what would it be? -5. This one's a little bit trickier. Well, this is a negative sign, but what about the number? Well, there's an implied 1, this is really -1. This one, well there's actually an implied 1 also. Well, yea, but you can think of this actually as one fourth, a quarter, times w. so, this ones a little tougher, but that is the answer, its a fourth.
Step 9: Algebraic Expression
An algebraic expression is any combination of variables and constants with mathematical operations. Addition, subtraction, multiplication, division, roots, and powers. Neither the equals sign nor inequality symbols are part of an expression.Here's some examples of algebraic expressions. As you can see, they are combinations of constants and variables with different mathematical operations. Now the following here are not algebraic expressions. Why? Well, the first one here has an equal sign. The second one is an inequality. A term is any part of an expression that is separated by plus signs.
Step 10: Example 1
For this example I want to list the terms of this expression right here. What are they? Well, they are separated buy plus signs. The first one is just three x squared. When you list them, you separate them with commas. The second term is 5x and the third one is just 7. Subtraction can always be rewritten as adding the opposite. So, for example, if you are given eight x cubed minus four x minus 1, you can rewrite that as follows. Eight x cubed plus a negative four x plus a negative one. So, to list the terms, all we have to do is this. Eight x cubed, negative 4 x, negative 1.Like terms, or similar terms, are terms that have the same variables raised to the same powers, but may have different coefficients.
Step 11: Example 2
Now lets take a look at some examples of what are like terms and what are not like terms. The first one, 3x and -7x, are like terms. Why? Well, there the same variable raised to the same power. The coefficients may be different. The next example, eight y squared , two y squared, and negative three y squared, all like terms. What about 5 and 1? Well, it turns out constants are considered like terms. Good to know.Here, its a little more complicated. We have x cubed y to the fourth. This is also x cubed y to the fourth. The coefficients are different. we have a 1 and 3/8, but thats fine. So These are also considered like terms.
Step 12: 6w and 8y
Now lets look at 6w and 8y. Are they like terms? Well no, because the variables are different. Over here, the variables are the same, but one is just 4 x to the first powerthere is four x to the second power. Its x squared. Those are not considered like terms. Again, the coefficient being the same is just trying to trick you. But I'm sure you didn't fall for that did you?Now its time for a Duh! definition of the lesson.Terms that are not like terms are called unlike terms.Duh! See? Told you.
Step 13: Terms of Algebraic Expression
Now lets identify all the like terms in the following algebraic expression. So with the 3y, theres nothing else that just has a y. Here's a y but its with an x squared.No like terms here. What about here? two x squared y, three x squared y, and hey, here's one at the end, So lets just list them. Two x squared y, three x squared y, negative x squared y. Here are the first set of like terms. Anything else? Negative x, seven x.Aha! So I can list those.
Step 14: Finally
Anything else? There's a four....You can think of that as negative five. So Aha! So, 4 , -5. Constants are considered like terms. Anything else? Nah, we've pretty much covered all our bases. So here is our list of like terms.
