In this lesson, our instructor Vincent Selhorst-Jones discusses Variables, Equations, & Algebra. After defining a variable, constant, and coefficient, he discusses expressions and equations. The Idea of Algebra, how to solve an equation, how to do order of operations, the distributive property, and substitution are also explored in great detail. Vincent finishes the lesson with four worked-out examples.
A variable is a placeholder for a number. It is a symbol that stands in for a number. There are generally two ways to use a variable:
The variable is allowed to vary. As its value changes, it will affect something else (the output of a function, a different variable, or some other thing).
The variable is a fixed value (or represents multiple possible fixed values) that we do not know (yet) or do not want to write out. Normally we can figure out the value by using information in the problem.
We normally use lowercase letters to denote variables, but occasionally we will use Greek letters or other symbols.
A constant is a fixed, unchanging number. Occasionally, we might use a symbol to refer to a constant (In such a case, we might refer to it as a variable, but we know that since it's a constant, the variable is fixed.).
A coefficient is a multiplicative factor applied to a variable.
An expression is a string of mathematical symbols that make sense used together. Often we will simplify an expression by converting it into something with the same value, but easier to understand (and usually shorter). For example, we might simplify
the expression 7+1+2 into the equivalent 10.
An equation is a statement that two expressions have the same value. We show this with the equals sign: =. For example, the equation
2x+7 = 47
says that the expression 2x+7 is equivalent (equal) to the expression 47. In other words, each side of the equation has the same value.
If we have an equation (or other kinds of relationships as well), we can do algebra. The idea of algebra is that since each side is equivalent to the other side, if we do the exact same operation to both sides, the results must also be equivalent.
This idea makes sense, but it's critically important to remember you must do the exact same thing to both sides when doing algebra. If you do different things on each side, you no longer have an equation. This is a common mistake, so don't let it happen
When you solve an equation, you are looking for what value(s) make(s) the equation true. Most often you will do this by isolating the variable on one side: whatever is then on the other side must be its value. You isolate the variable by
doing algebra. Ask yourself, "What operation would help get this variable alone?", then apply that operation to both sides.
It's critical to remember the order of operations when simplifying expressions and doing algebra. Certain operations take precedence over others. In order, it goes
Parentheses (things in parentheses go first),
Exponents and Roots,
Multiplication and Division,
Addition and Subtraction.
Distribution allows multiplication to act over parentheses. The number multiplying the parentheses multiplies each term inside the parentheses:
3(5 + k + 7) = 3·5 + 3k + 3·7.
We can also use the distributive property in reverse to "pull out" a factor that appears in multiple terms:
3x2 + 7x2 − 5x2 = (3 + 7 − 5) x2.
We can use information from one equation in another equation through substitution. If we know that two things are equal to each other, we can substitute one for the other.
x = 2z + 3, 5y = x−2 ⇒ 5y = (2z+3) − 2.
When we substitute, we need to treat the replacement the exact same way we treated what was initially there. The best way to do this is to always put your substitution in parentheses.
Variables, Equations, & Algebra
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.