Section 2.1 Letters are Numbers in Disguise
Learning Objectives
Substitute a number for a variable and then simplify the expression.
Manipulate linear equations with multiple variables.
Some people like to say that math made sense when it was all numbers, but then things fell apart when the letters started showing up. This is unfortunate, because the letters are actually just a way of representing numbers, and thinking about them in that way can help to clarify confusion. There are a lot of ways that teachers talk about variables and they are of varying degrees of correctness. For example, some teachers say that variables represent unknown values. It's not exactly wrong, but it's not exactly right. Whether or not we know the value is irrelevant. In college level math courses, we can use variables to represent mathematical expressions with other variables in it instead of just thinking of it as a quantity. This leads us to a more general definition:
Definition 2.1.1. Variable.
A variable is a symbol that represents a quantity or a mathematical expression.
Notice that we allow for variables to represent both quantities and mathematical expressions. What this means is that both \(x = 4\) and \(x = y + 1\) are valid uses of variables.
Activity 2.1.1. Variables as Quantities in Expressions.
What does it mean for a variable to represent a quantity in an expression? One way to think about it is that it's a calculation waiting to happen. Consider the expression \(2x\text{.}\) This represents a number, but which number it represents depends on the value of \(x\text{.}\) As soon as I give you a value for \(x\text{,}\) you can plug it in and give me a specific number. But until then, this is just a number that is waiting to be calculated. For example, if you know that \(x = 4\text{,}\) then you can calculate that \(2x = 8\text{.}\) We can write this as an English sentence:
Notice that we're not saying that \(2x\) always has the value of 8. We're just saying that if \(x\) takes the value of 4, then the value of \(2x\) must be 8.
Try it!
Determine the value of \(2x\) when \(x = 5\text{,}\) \(x = 12\text{,}\) and \(x = -3\text{.}\) Write your results as if-then statements.
Solution.Activity 2.1.2. Variables Represent Quantities (Part 1).
Consider the following set of equations:
Notice that to solve all of them, you would subtract 3 from both sides of the equation. This algebraic step solves the equation regardless of what number is on the right side of the equation.
Now consider this equation: \(x + 3 = a\text{.}\) We will solve this variable for \(x\text{.}\) Notice that the algebraic step remains the same. It turns out that the final result cannot be simplified any further.
Try it!
Solve the equation \(x - 7 = c\) for the variable \(x\) using a complete presentation.
Solution.Activity 2.1.3. Variables Represent Quantities (Part 2).
Consider the following set of equations:
This time, the number that is being added to \(x\) is changing. But these are still very similar equations. And the algebraic step is basically the same for each one. You need to subtract whatever number it is that's being added to the \(x\text{:}\)
Now consider this equation: \(x + b = 7\text{.}\) Based on the pattern above, you should have an idea of what you need to do to solve for \(x\text{.}\)
Try it!
Solve the equation \(x + b = 7\) for the variable \(\) using a complete presentation.
Solution.Activity 2.1.4. Using Other Symbols.
The variable \(x\) is traditional, but not special. We're allowed to solve for other variables. The underlying thinking process remains the same.
Try it!
Solve the equation \(a + b - c = d - e\) for the variable \(c\) using a complete presentation.
Solution.