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Algebra Study Guide | Examples and Practice Problems

Algebra Study Guide | Examples and Practice Problems

Welcome to the realm of algebraic examples, where equations unfold and variables align. We get that algebra can sometimes seem like a puzzling maze, but fear not, for we’re here to guide you through the twists and turns. Now that we’ve squared away the formal definitions, prepare to explore the art of simplifying expressions, solving equations, and unraveling the mysteries of quadratic functions. We’ll help you overcome any confusion and empower you with the tools to succeed.

Let’s get learning! But before we dive in, here’s a handy reminder you’ll want to bookmark.

The three rules of algebra:

  1. 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.
  2. 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).
  3. 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.

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 any algebraic equation, you can use your Photomath app to scan the problem (via mobile phone camera) and get guided through a step-by-step explanation.

handwritten algebra problemscan

Algebra Equations

Okay, we’re ready to explore some real-life examples of algebra now, let’s take it one step at a time.

Example 1

We’ll solve this linear equation:


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


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 $$-3x$$:


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:


Again, collect like terms:


Calculate the difference:


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 together:


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:



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

$$x_1=-5, x_2=5$$

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 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 Photomath 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.