Note: You can skip this lesson if you are already familiar with basic algebra concepts.

This lesson is designed to help you review basic algebra concepts that are necessary for understanding physics. This lesson will cover the basics of algebraic operations, equations, and functions that will be occasionally used throughout the course, even in the conceptual difficulty.

Introduction

To start off, let's begin by talking about what is algebra. Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols.

Why is this important in physics? Well, the physics equations that you are about to learn aren't just simply "equations". They actually describe a relationship between different physical quantities, for example, let's take a look at Newton's Second Law of Motion: $$F = ma$$ It may seem arbitrary now, but this equation actually describes the relationship between force, mass, and acceleration. In this case, the letter "F" represents force, "m" represents mass, and "a" represents acceleration. Thus, the equation tells us that the (net) force acting on an object is equivalent to the mass multiplied by the acceleration.

Variables

Let's talk the very core of algebra: variables. Variables are symbols or letters that represent quantities, especially unknown ones. All the letters in the previous equation are variables, as $F$ stands for force, $m$ stands for mass, and $a$ stands for acceleration. Variables can hold numerical values too, like this: $a=5$. When we declare this, we are saying that $a$ is equal to $5$, so that means wherever we see and $a$ in an equation or expression, we can replace it with $5$. Going back to the previous example, if we know that $F=ma$ and we know that mass is $2$ and acceleration is $5$ (i.e. $m=2$ and $a=5$), then we can replace the variables with their values to get: $$F = 2 \times 5$$ And we can solve this to get $F=10$. This method is called substitution.
Obviously we can't just randomly assign value to variables most of the time, usually they are given or we have to solve for them. But this is the basic idea of variables in algebra. They are just letters that represent quantities, and we can manipulate them to solve equations or expressions.

Syntax

First, we need to dissect the way these equations are written, which may be unfamiliar to you. Here is the equation again: $$F = ma$$ An equation of letters and symbols is no different from an equation of numbers, like $4+4=8$. They both indicate a relationship between multiple quantities, just that we replaced the numbers with letters. Furthermore, you might know that "$\times$" is the multiplication operator, but in algebra, we can actually use "$\cdot$" or just put two letters next to each other to indicate multiplication. In essence, the above equation is the same as saying, $$F = m \times a$$ We don't typically use the multiplication operator in algebra because it looks like the letter $x$ and can cause confusion.

Coefficients are also good to know and recognize. A coefficient is the number in front of a variable that you might see, like $2a$. This is the same thing as $2 \cdot a$, or $a+a$. Think of it like how you learned multiplication in elementary school: $4 \cdot 4 = 4+4+4+4$. A coefficient works the exact same way.

You might also see us using $\Sigma$ and $\Delta$ here and there, just note that these are NOT variables and should NOT be treated like variables. Instead they are used as clarifiers, as $\Sigma$ represents "sum", or "net", like $\Sigma F$ represents net force, and $\Delta$ means change in, like $\Delta x$ means change in $x$. Again, they are not variables, so treat it and the letter/symbol that follows as a single variable. You might also see other Greek letters being used, like $\theta , \pi , \alpha , \tau , \omega$. Just note that this time, these will actually refer to variables, and not whatever $\Sigma$ and $\Delta$ refer to. It might be a bit confusing at first, but it will become easier to understand later on.

Another important notation you might want to know is exponents. For example, you might see us use $v^2$. This is simply equal to $v \cdot v$. Likewise, for $v^3$, it would be $v \cdot v \cdot v$. Sensing a pattern? I'll leave it to you to figure out what $v^4$ and $v^{100}$ mean then.

On the topic of exponents, you might also see us using subscripts, like $v_i$ and $v_f$. The reason we use these is very similar the function of $\Sigma$ and $\Delta$, the subscript is simply for more clarification. For example, $v_i$ stands for initial velocity, and the latter stands for final velocity. Sometimes we might use numbers as subscripts; they serve the same purpose: $r_1$, $r_2$, and $r_3$ refer to three different variables.

Basic Algebraic Operations

Next, we will talk about some basic algebraic operations that will help you understand how we can manipulate these equations and expressions to our benefit. The most basic operation is addition/subtraction. We can add or subtract quantities to both sides of the equation without changing the relationship. For example, going back to our original equation, $F=ma$, we can add say, $5$ or even a random variable $x$ to both sides of the equation like so: $$F+x=ma+x$$ ...And the equation will still hold true. Note how you have to do this to both sides of the equation, not just one. Otherwise, the equation won't be true anymore. Think of it with numbers if it doesn't make sense: If we have $2 \cdot 2 = 4$, then if we only add $5$ to one side for example, both sides won't be equal anymore. The principle here is the exact same, we are using variables instead of numbers.

The same can be done with multiplication and division (as long as you don't divide by zero). We can multiply or divide both sides of the equation by a quantity without changing the relationship. For example, If we multiply $F=ma$ by $2$, we get: $$2F = 2ma$$ And the equation still holds true. Division works exactly the same way. If this isn't clear, again, we can see this through a numerical example: $2 \cdot 2 = 4$, multiplying both sides by $2$ gives us $2 \cdot 2 \cdot 2 = 4 \cdot 2$, which is $8=8$.

Finally, we can use these operations to our advantage when we want to solve for a variable. For example, we can solve for $a$ in the previous equation by dividing both sides by $m$: $$\frac{F}{m} = \frac{ma}{m}$$ $$\frac{F}{m} = \frac{m}{m} \cdot a$$ $$\frac{F}{m} = a$$ (Recall that anything divided by itself equals $1$, and $1$ multiplied by anything is itself) This is the same as saying that acceleration is equal to force divided by mass. We can do this with any variable in an equation, as long as we follow the rules of algebra and manipulate the equation correctly like we just showed you.

We can also actually add equations together too. For example, if we these two equations: $$x+y=5$$ $$2x+y=7$$ We can add them together, by adding the left sides of both equations together and the right sides of both together to achieve: $$x+y+2x+y=5+7$$ $$3x+2y=12$$ However, this equation isn't all that helpful to us. It might be helpful to know that we can actually subtract two equations as well: $$2x+y - (x+y) = 7 - 5$$ $$2x-x+y-y=2$$ $$x=2$$ ...And you can actually see how we used these two equations to solve for $x$! Now a challenge to you: Can you figure out how to use this new information about $x$ to solve for $y$? Try it out and see!

Let's work through an example problem together to solidify your understanding of these concepts.

Now that you have a solid understanding of basic algebra, it's time to move on to your first lesson in physics! (You can check out the practice problems in this lesson if you feel like you need more practice with algebra)