# Introduction to Algebra – Basics, Concepts, Examples, Practice

Ah, algebra… This is a real milestone in a math student’s journey. One day you’re practicing times tables, and the next? There are letters now?!

But there’s no need to panic (really!). We’re here to make sure that, with a little background information and a willingness to learn, you can embark on this algebraic adventure calmly and confidently.

Roll your shoulders back, unclench your jaw, and let’s get started!

## So, what is algebra?

Essentially, algebra is operations and relationships of variables.

Basic operations are what you already know and love – addition, subtraction, multiplication, division – but there are also more complicated operations like exponentiation, taking roots, and more.

Relations are just the connections between two elements, whatever they are. For example, two elements can be equal, one can be greater than the other, etc. — those are all relations!

A variable is a symbol with no fixed value. You’ll see variables pop up in equations as anything from $$a$$, $$b$$, or $$c$$ to $$x, y$$, and $$z$$; but, because they’re letters, they have no fixed value (e.g. “$$x$$” is not always equal to the same number; sometimes, it’s not even equal to anything at all). Solving the mystery of these variables is what makes algebra so fun!

Think of it this way: Instead of $$2 + 2 = 4$$, you’d see $$2 + 2x = 4$$ and have to figure out what value that variable $$x$$ could represent.

It’s important to know that algebra is an essential building block for calculus, trigonometry, and other branches of math, so make sure you really take the time to understand these fundamental concepts.

Believe it or not, algebra will become that extra-comfy sweatshirt that you reach for over and over again when you’re out of ideas and don’t know where to start. If you start to wonder why you should care, repeat this to yourself: confidence in algebra will mean comfort down the road!

### The definition of algebra

If you’re looking for a more formal definition of what makes algebra algebra, here’s our favorite basic definition:

**Algebra** | *noun*. the branch of mathematics that describes operations and relations between different types of mathematical objects (sets of numbers, values, vectors, etc.) in the most general way

This definition is comprehensive but not too complicated, so it’s a great home base for defining this branch of math.

### Basic algebra

Before we get into examples or practice problems, let’s break down the basics!

First, let’s look at some vocabulary terms:

Key Term | Definition | Example |
---|---|---|

Variable | An entity with unknown value | $$x, y, z$$ |

Operator | Denotes the operation | Addition, multiplication, etc. |

Constant | An entity with a fixed value | $$3, 200, \frac{1}{2}, π$$ |

Coefficient | The constant attached to a variable | $$\text{ The } 3 \text{ in } 3x$$ |

You’ll hear those terms used during lessons and in textbooks, so that’s a good list to keep in your back pocket!

Basic algebra might start with only one variable, but as you progress on your journey, you’ll see more variables (and more operators!) which will make your value sleuthing a little more difficult.

## What is algebra used for?

Algebra is used for so much more than just homework! Algebra is employed to uncover an unknown value, which is helpful in everything from calculus class to economic analyses. If you’re trying to calculate how many hot dogs you started with if you ate $$2$$ and there are $$3$$ left… guess what? That’s algebra! You’re solving $$x – 2 = 3$$, and you didn’t even realize it!

## Algebra concepts

Algebra is an expansive branch of mathematics that encompasses many different concepts, like:

- Algebraic expression vs. algebraic equation
- Combining like terms
- Compound inequalities
- Substitution and evaluating expressions
- Linear equations
- Slope
- Systems of equations
- Evaluating functions

… and more!

Depending on your grade level or class, you might not know some of those concepts yet, which is totally okay. You’re still learning, and that’s amazing!

Whether you’re in pre-algebra or algebra 2, there are certain consistencies and properties that can help you navigate through problems.

If you ask us, this is one of the coolest parts of learning algebra. These properties are like changing your perspective – the information stays the same, but the way you see it makes things clearer!

Basic algebraic properties | Property |
---|---|

Commutative Property of Addition | $$a + b = b + a$$ |

Commutative Property of Multiplication | $$a × b = b × a$$ |

Associative Property of Addition | $$a + (b + c) = (a + b) + c$$ |

Associative Property of Multiplication | $$a × (b × c) = (a × b) × c$$ |

Distributive Property | $$a × (b + c) = (a × b) + (a × c), ~a × (b - c) = (a × b) - (a × c)$$ |

Additive Identity Property | $$a + 0 = 0 + a = a$$ |

Multiplicative Identity Property | $$a × 1 = 1 × a = a$$ |

Additive Inverse | $$a + (-a) = 0$$ |

Reciprocal | $$\text{Reciprocal of } a \text{ is } \frac{1}{a}$$ |

Algebraic property names can sound complicated or overwhelming, but if you spend enough time with the column on the right, you’ll see it’s all pretty logical and not as scary as you think.

## Understanding algebra

Understanding algebra is possible – we promise! In addition to understanding concrete things like the definition of algebra, vocabulary terms, and algebraic properties, the following mindset suggestions can help you unlock the door of truly understanding algebra:

- Stay curious: After all, algebra is about solving an unknown.
- Don’t look at it like a chore or a bore; instead, try maintaining a genuine curiosity about how to find your secret value. You don’t have to go full Sherlock Holmes, but you can still have fun with it!
- Take a step back: It can be easy to get lost in a jumble of letters and numbers. It’s okay to close your eyes for a moment, take a deep breath, and start fresh.
- Respect the process: Algebra teaches us to focus on the “how” when we’re faced with unknowns; remembering to appreciate the process can be helpful with your homework and beyond!

## What is an expression in algebra?

Simply, an algebraic expression is a combination of terms joined by operation, like $$3x + 8$$. An expression can include variables, constants, coefficients, and operators.

An algebraic expression becomes an algebraic equation with the inclusion of an equal sign ($$=$$), like $$3x + 8 = 24$$.

## Examples of algebra

Okay, now that we’ve got all of the formal definitions out of the way, let’s look at some real examples.

### Example 1

We’ll solve this linear equation:

First, we’ll apply the distributive property, meaning we’ll distribute $$2$$ through the parentheses:

$$4+2x=3-x-3x$$

Next, we’ll collect the like terms. Like terms are the terms that have the same variables raised to the same power. In this case, the like terms we’ll collect are $$-x$$ and $$-3x$$:

$$4+2x=3-4x$$

The goal of solving an equation is to isolate the variable on one side of the equation and leave all the constants on the other. So, let’s do that: Move $$-4x$$ to the left-hand side and change its sign. At the same time, move $$4$$ to the right-hand side and change its sign:

$$2x+4x=3-4$$

Again, collect like terms:

$$6x=3-4$$

Calculate the difference:

$$6x=-1$$

To get the final solution, simply divide both sides by $$6$$, the coefficient next to the variable:

### Example 2

Now, let’s solve this quadratic equation:

Notice that, for $$x$$ to be isolated on the left-hand side, we need to get rid of the exponent $$2$$. We’ll do that by applying something that is opposite to squaring the variable – taking the square root! But we must be careful! We have to take both positive and negative roots:

Why did we have to take both roots? The answer is simple: both $$5$$ and $$-5$$, when squared, give the same result: $$25$$. Now let’s separate the solutions:

$$x=-5$$

$$x=5$$

That’s it! We got our results! We can index them if we want to:

Of course, these are just a few examples of algebra problems. Remember that if you need help with a specific problem, you can always use your Photomath app to scan the expression or equation and bring up detailed, expert-verified explanations.

## Algebra 1

Let’s get a little more specific. If you’re wondering what is usually taught in an Algebra 1 class, here are the topics generally covered in this first level, sometimes called elementary algebra:

- Evaluating inequalities and solving equations
- Inverse operations
- Distributive and commutative properties
- Polynomials
- Linear equations
- Quadratic equations
- Graphing lines and parabolas

Students will typically come across these Algebra 1 topics in eighth or ninth grade, but some math learners may work at a different pace.

## Algebra 2

As expected, Algebra 2 gets a little more complicated – or, as we like to say, a little more fun! If you’re an Algebra 2 student, you’re probably learning:

- Graphing functions and linear equations
- Matrices
- Polynomials and radical expressions
- Quadratic functions and inequalities
- Exponential and logarithmic functions
- Sequence and series

If you took Algebra 1 in eighth or ninth grade, you’ll probably take Algebra 2 in tenth or eleventh grade; however, all paces are welcome here!

## Algebra 1 vs Algebra 2

Basically, Algebra 1 is all about setting you up for success in Algebra 2 and beyond. Algebra 1 is heavy on rules, terms, formulae, and methods so that you have a tool chest of information when you encounter more complicated problems in Algebra 2, calculus, trigonometry, or even geometry.

## Solving algebraic equations

Let’s touch on actually solving algebraic equations! Remember that this is very process-oriented, so don’t be afraid to lean on rules, properties, and formulae that apply to your problem.

### How to solve algebraic equations

How to solve an algebraic equation will vary depending on your specific problem, but your goal is typically to isolate your variable on one side of the equation in order to clearly find its value.

You can do this in many ways, depending on the operators in your equation, but remember that whatever you do to one side of the equation must also be done on the other side!

For specific help with an algebraic equation, use your Photomath app to scan the problem and get guided through a step-by-step explanation.

## Algebra practice problems

Now that we have a thorough understanding of what algebra is, why we use it, and how we solve it, we can push that initial fear of the unknown aside and get into some practice problems!

Practice is how we learn and grow – and yes, that includes getting it wrong sometimes. That’s more than okay! In fact, it’s normal and necessary to stumble a few times during practice. Remember that as we try to solve the following:

Problem | Solution |
---|---|

$$3x+5=17$$ | $$x=4$$ |

$$5=0.5y-4$$ | $$y=18$$ |

$$9+\frac{1}{4} x=1$$ | $$x=-32$$ |

$$3(x+1)-5x+8=9$$ | $$x=1$$ |

$$x+1<15$$ | $$x<14$$ |

$$y^2=16$$ | $$y_1=-4,y_2=4$$ |

$$a^2+2a+1=4$$ | $$a_1=-3, a_2=1$$ |

$$x^3=8$$ | $$x=2$$ |

$$(x-1)(x+2)\leq0$$ | $$x∈[-2,1]$$ |

**Here’s how we solve the first problem in the app:**

Want to check your work or see a different problem? Use your Photomath app to scan and check your solving steps against our teacher-approved methods.

**FAQ**

*What are the 3 rules of algebra?*

The three rules of algebra are:

- The Commutative Property (for addition and multiplication): This allows you to reorder your elements on either side of a plus sign or multiplication sign without changing the result. For example, $$1 + 2$$ equals the same sum as $$2 + 1$$; likewise, $$3 × 4$$ gets you the same product as $$4 × 3$$.
- The Associative Property (for addition and multiplication): This property enables you to move parentheses around elements of an expression when the operators are all addition or all multiplication – and still get the same result. Here’s an example: $$(1 + 2) + 3$$ is the same result as $$1 + (2 + 3)$$. By the same token, $$(4 × 5) × 6$$ will have the same result as $$4 × (5 × 6)$$.
- The Distributive Property (for addition and multiplication): The distributive property is about distributing a coefficient into a parenthetical expression, like $$3(a +b)$$ or $$2(c – d)$$, where the coefficient attaches itself to the elements inside the parentheses and we keep the operator. In these examples, that means $$3(a + b) = 3a + 3b$$, and $$2(c – d) = 2c – 2d$$.

*What are five different types of algebra?*

Although others may define them slightly differently, we like to say the five types of algebra are:

- Elementary algebra – introduces algebraic quantities (real and complex numbers, variables and constants); rules of operations for those quantities; geometric representations; formation of expressions and sentences (equations, inequalities); rules of manipulation with expressions and equations; and how to solve algebraic equations and systems of equations.
- Abstract algebra – a study of algebraic structures along with their associated homomorphism. Algebraic structures include groups, rings, fields, modules, and more!
- Advanced algebra – the branch of algebra that connects algebraic concepts with other math fields like trigonometry and calculus
- Commutative algebra – the branch of algebra that studies commutative rings as part of algebraic number theory and algebraic geometry
- Linear algebra – a study of vector spaces and linear transformations