Algebra

Variables and Equations

Often in math, we have to deal with problems that force us to find what number something is. For example, say Alice and Bob have a combined total of 10 apples. We know Alice has 6 apples, so how much does Bob have? Our goal is to find the number of apples Bob has. Let’s talk about this scenario in a math way:

\[\text{Number of apples Alice has} + \text{Number of apples Bob has} = 10\]

Because Alice has 6 apples, Bob must have four to make this equation work:

\[6 + \text{Number of apples Bob has} = 10\]

It’s important to notice that the Number of apples Bob has is an unknown, because the question does not give the straight number - we have to solve the problem to figure that out. We call these unknowns variables. Variables just represent an unknown number.

Because writing out “Number of apples Bob has” can get tedious, let’s just shorten that to “b”, for Bob. \(b\) is a variable. So, we can model the question in the following way:

\[6 + b = 10 \\ \text{Solve for }b\]

Here, \(6 + b = 10\) is an equation. An equation is simply two expressions with an equal sign in between. Here, our expressions are \(6 + b\) and \(10\). Expressions are simply a set of numbers or variables that interact with each other using operations. Operations are things like addition, subtraction, multiplication, and division. Important note: whenever you see a number next to a variable, that signifies multiplication. For example, \(6x = 6\cdot5 = 6\) x \(5\).

Another important thing to know is what a constant is. A constant is simply a number. In our equation, \(6\) and \(10\) are constants.