Letters in Math?! Introduction to Algebra

Prerequisites Required- None

Before coming to this blog (and having a sufficient reading maturity to understand the content), I assume you have learned the fundamentals of arithmetic (addition, subtraction, multiplication, and division of numbers). $4+4 = 8$, $10*6 = 60$, $7-3 = 4$, and $\frac{12}{2} = 6$ should all be very familiar to you.

To exist in our society requires arithmetic skill. When it’s 11:37 and you calculate there are 23 minutes until lunch at noon you have used arithmetic. When someone hands you a five dollar bill for a $\$4.75$ churro, you hand them $\$0.25$ back and have done arithmetic.

Algebra is the next step after arithmetic and is typically taught to students entering their teen years. It is a more abstract form of mathematics centered around undefined numbers called variables. Variables are typically represented by english or greek letters. $x,y,z,a,b,c,\theta,\alpha,\beta,\gamma$ are all common symbols used to represent variables.

Unlike arithmetic, algebra isn't strictly required to function as a human being in society and I have found that people who don't use it regularly need a refresher. I hope this brief summary will spark some memories (if you have already learned) or motivate your current study of the subject.

Formulas

Suppose I make the statement $a=2$. I've now assigned the variable $a$ a value of 2. Whenever I talk about $a$, you know that it has a value of 2 and I can use it in any context. What is $5 + a$?

$$5 + a = 5 + 2 = 7$$

This applies for the other arithmetic operations (subtraction, multiplication, and division):

$$9 - a = 9 - 2 = 7$$

$$6*a = 6*2 = 12$$

$$\frac{8}{a} = \frac{8}{2} = 4$$

We can also use multiple symbols in an equation. Suppose $a = 2$ and $b = 3$:

$$a + b + 1 = 2 + 3 + 1 = 6$$

The use of variables allows us to define relationships without needing to define the values. For example, suppose that we have a rectangle with two sides of length $L_1$ and $L_2$. The area of the rectangle, represented by the symbol $A$, is:

$$A = L_1*L_2 \:[1]$$

This is called a formula. The $[1]$ at the right of the equation indicates that this is equation number $1$ so that we can refer to it in paragraph format text. Formulas are ubiquitous in engineering; they represent a mathematical machine that accepts inputs and returns an output. Equation 1 returns the area of a rectangle from the length of two sides. Note that sometimes, to make things cleaner looking, we will omit the multiplication sign and smash the letters right next to each other. This represents multiplication.

$$A = L_1L_2$$

We can also smash letters together with parentheses around them to represent multiplication:

$$A = (L_1)(L_2) = L_1(L_2) = (L_1)L_2$$

Addition, subtraction, and division will always use a specific symbol of some kind to represent those operations.

We can rearrange the mathematical machine to change the inputs and outputs. For example, if we divide both sides of the equation for the area of the rectangle by $L_2$, we can obtain the remaining side length given the area and the length of the other side:

$$\frac{A}{L_2} = \frac{L_1*L_2}{L_2} $$

$$\frac{A}{L_2} = L_1 \: [2]$$

In general, we can add, subtract, multiply, or divide an equation by any number or variable on both sides without fundamentally altering it. For example, we could multiply equation 2 by $L_2$ to get back the previous formula:

$$\frac{A}{L_2}L_2 = L_1*L_2$$

$$A = L_1*L_2$$

Solving for Variables

Another power of algebra is to solve for the value of a variable. Consider the equation:

$$5 + x = 7\:[3]$$

Let’s plug in a random value of $x$ and see if the equation is valid, $x = 3$:

$$5 + 3 = 7?$$

This statement is clearly invalid. How about $x = 4$?

$$5 + 4 = 7?$$

Also wrong. We will likely eventually guess correctly, but it could take a while. We could instead use algebra to solve for $x$. Remember that we can

• Add to both sides of the equation.
• Subtract to both sides of the equation.
• Multiply both sides of the equation.
• Divide both side of the equation.

For example, we could multiply both sides of equation 3 by 3 to obtain:

$$3(5+x) = (3)7$$

Due to the distributive property of arithmetic, $3(5+x) = 3(5) + 3(x)$ so we obtain:

$$15 + 3x = 21\:[4]$$

This doesn’t really seem to have gotten us very far. Let’s instead try subtracting 5 from both sides of equation 3:

$$(5 + x) - 5 = 7 - 5$$

$$x = 2$$

Huzzah! Let's substitute the value we calculated to verify:

$$5 + x = 5 + 2 = 7$$

As another verification, if we set $x=2$ in equation 4, the equation is valid, because after we multiplied both sides by $3$, equation 4 is still inherently the same as equation 3:

$$15 + 3(2) = 21$$

This demonstrates a very important fact. Algebraic operations do not change the fundamental relationship of the variables in the equation.

Linear Equations

What happens if we have an equation with two variables:

$$y = x + 2$$

Let’s try the subtraction trick from earlier:

$$x + 2 - 2 = y - 2$$

$$x = y - 2$$

Well… this isn’t any better.  Let’s go back to randomly guessing values for $x$:

$$x = 2$$

$$2 + 2 = y$$

$$y = 4$$

Let’s try again with, $x = 5$:

$$5 + 2 = y$$

$$y = 7$$

Different values for $x$ have yielded different values for $y$. Unlike in the 'solving for variables' section where there was only one value for $x$ that made the equation true, here, we can input any value for $x$ and obtain a corresponding value for $y$ that makes the equation true. This is because there are two variables and only one equation (before, there was one variable and one equation). This equation is called a function.

Functions, similar to formula, are mathematical machines that accept inputs and return an output, but instead, we are more interested in the relationships between the inputs and outputs rather than any specific computation.

Functions are typically written as either an equation of two variables, $y = 4x + 2$ or with a notation called function notation, $f(x) = 4x + 2$. A person reading $f(x)$ would hear in their mind "f of x", that is, the function is called $f$ and the input variable is $x$.

The function $f(x) = 4x + 2$ is called linear because it is a function of the form $f(x) = mx + b$ where $m$ and $b$ are any numbers. $f(x) = 3x + 12$ is also a linear function.

Quadratic Functions

Functional relationships can take just about any form. A quadratic function is one where $f(x) = ax^2 + bx + c$. For example, $f(x) = 4x^2 + 3x + 1$ or $f(x) = 7x^2$ are both quadratic functions.

Final Thoughts

There are many more kinds of functions than linear and quadratic, but these two kinds are very common.

Plugging values into formulas, variable solving, and linear/quadratic functions are amongst the most common algebraic concepts in engineering.