Algebra: 100 Fully Solved Equations To Explain Everything

Book Preface

You are about to embark on a journey into mathematics, so don’t freak out. More specifically, this journey is all about algebra, which is basically one of the major divisions of math. While your experience with algebra may have so far been confusing, this book is designed to make it simpler for you. Algebra is, in fact, a way to look at math problems that actually makes math problems simpler. This is especially true of word problems, which can be converted easily into basic algebra problems that you will be able to solve after reading this.
The key thing to know about algebra is that is based not only on numbers but on things called “variables”. What is a variable? A variable is something about the problem that you don’t know the value of. In fact, algebra can be considered a type of mathematics that is concerned with the solving or determination of what number value the variable has. In algebra, mathematicians use placeholders, such as “x”, “y”, and “z” to indicate that these represent numbers of which we do not yet know the value. Your job is to know what the value of the variable is based on the rest of the known numbers in the equation.
The rest of algebra is basically the same arithmetic you’ve known since grade school. Addition is still addition, subtraction is still subtraction, and so on. Algebra also uses things like the square of something and the square root of something, which may not be something you truly understood in middle school or high school. The square of something is nothing more than the number multiplied by itself. This would be written like this: Two squared or 22 is the same as saying 2 x 2 or 4. It is written as 22 = 4. In the same way, the cube of something is the number multiplied by itself three times. 23 = 2 x 2 x 2 = 8. The square root is the reverse of this. The square root or √ of something is the reverse answer to the square of something. For example, the square root of 4 or √4 is 2 because 2 x 2 = 4.
The other thing you’ll have to remember and do as part of algebra involves the idea that equations are always balanced. In other words, the equation 2+ 2 = 4 means that everything on the left side of the equation or “=” sign is the same as everything on the right side of the equation. And, if you’ll think back to grade school, you can manipulate these equations. You can take 2 + 2 = 4 and turn it into 2 = 4 – 2. Easy, right? Well, in algebra, you’ll see things like x + 2 = 4, and you’ll have to “solve for x”, meaning that you have to put x on one side of the equation and figure out what x means. It looks like this: x = 4 – 2. The answer is, of course, “2”. This is algebra. While there are harder problems than this, you can use these basic rules to solve equations that, in this book, will get harder as we progress.
To be clear about the nomenclature, we will use x, y, z, a, b, and c to represent the variables and it will be up to you to determine the number the variable represents. There will also be some differences in how things are represented in print. In order to avoid confusion, we’ll use these symbols to represent mathematical equations:

Addition. This will always involve the “+” sign you are used to seeing.
Subtraction. This will always involve the “–” sign you are used to seeing.
Multiplication. This will be written in one of four ways. The “*” sign will be used in some equations, such as 2 * 6 = 12. More often, however, you will see it this way: 2(6) = 12 with the parentheses used to represent multiplication with the number next to it. It may also be written like this: (2)(6) = 12. This also represents 2 multiplied by 6. This allows for equations like this: 2(2+4) = 12. In this situation, the mathematical protocol is to do what’s in the parenthesis first and then do the mathematical problem of multiplication next. Why do this?Because 2 * 2 + 4 is completely ambiguous. 2 * 2 = 4. Add 4 to that and you get 8. BUT, 2 * (2+4) is the same as 2 * 6 and this answer is 12, not 8. The parenthesis tells you which set of operators to do first. Finally, you might see 2x or 3y. The combination of the integer and the variable together like this means you are supposed to multiply them.
Division. This is not represented by the ÷ sign (although it COULD be used). Instead the “/” symbol is used so that 2/4 = two-fourths or one-half. In the same way 4/(2+2) is 4/4 or 1. In algebra, you might see 2/x or (3 -2)/y. This is no different than you see when you’re just using integers.
Squared or cubed. This will be as you know it. Squared is x2 and cubed is x3 and so on. Know that it is possible to have any number other than 2 or 3, such as xn in which n represents any possible number.
Square root. The symbol “√” will be used to represent the square root of something. It will be possible to have the equation √(2+2), in which the entirety of 2 + 2 is under the square root. Again, the mathematical operators inside the equation are done first and then the square root.

So, in this book, we will look at easy, medium, and hard questions involving algebraic equations. Not only will we solve them together but we will work together to see how these equations get solved. So, if you’re ready, we’ll get started.