EQUATIONS & INEQUALITIES | TUTORIAL | MATH

Understanding and Solving Equations, a Complete Breakdown

Learn and understand how to solve and avoid common mistakes when tackling mathematical equations.

Matt Connors

5 min readDec 5, 2024

Today, we’ll tackle mathematical equations, a topic that has puzzled me for a while.

By exploring the basics and breaking it down step by step, I hope to clarify it for both you, the reader, and myself.

Why Solving Equations is Important

From finance to programming and beyond, equations are a foundational concept in almost all fields that even slightly revolve around math.

Being able to not only solve equations but also understand them can help you, unlocking lots of potential doors for the future.

I have struggled to understand this concept for a while, so I figured I could try and explain it here and maybe understand it better myself.

With that in mind, let’s break down what equations are and how they work.

What is an Equation?

You can think of an equation like a scale with two parts with an = in the middle.

To balance a scale, both sides’ values must be equal, but what they are actually made from doesn’t ultimately matter.

Think of that scale holding two separate stacks of items: on one side, the scale holds 1lb of feathers, and on the other side, the scale holds 1lb of rocks.

Even though the items themselves aren’t the exact same — we have yet to live on a planet with feathery rocks — the values of the items, one pound, remains the same, so the scale is in balance.

In an equation, both sides must be exactly equal to each other in value, but that value can be represented in different ways, just like how different languages have different words for the same thing.

In summary, an equation is like a scale: the values on both sides must remain equal. These values might look different but hold the same value, just like how 1lb of feathers equals 1lb of rocks.

This idea of balance is the foundation of solving equations.

The Basic Rules of Equation Solving

To “solve” an equation essentially means to find its simplest possible form, or to specifically find the value of one of the terms contained therein by isolating it.

There are a few rules for solving equations which you need to keep in mind.

1. Any operation that is done to one side of the equation must also be done to the other side.

x + 5 = 20 is an equation. To find the value — or solve for — x, we need to subtract 5 (the neighboring value of x) from both sides of the equation.
x + 5 = 20 | — 5 ⇒ x + 5–5 = 20–5. As we know, + and — cancel out if the number afterwards is the same, so +5–5 becomes 0.
x + 5 — 5 = 20–5 ⇒ x = 20–5 ⇒ x = 15. So, x is 15. We can also say that the solution of the equation is 15.

2. You must solve an equation that contains multiple operations in the reverse order that it was written in — given that PEMDAS gave you the equation, you need to go backwards in order to undo the operations.

2x — 4 = 10 is an equation. To solve for x, we need to go backwards of PEMDAS. So, while you might think that we need to start by cancelling out the multiplication of 2x, we actually need to cancel out the -4 first.
2x — 4 = 10 | +4 ⇒ 2x = 14. Now that we got to the multiplication, we need to divide both sides by 2.
2x = 14 | / 2 ⇒ 2x / 2 = 14/2 ⇒ x = 7

For those not in the know, PEMDAS is an acronym for the order of operations: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction.

So, to reiterate, to solve 2x — 4 = 10:

Start with the addition/subtraction:
2x — 4 + 4 = 10 + 4 => 2x = 14
Then handle the multiplication/division:
2x / 2 = 14 / 2 => x = 7

Moving Terms to the Other Side

Once again, there are two sides to an equation, the left and the right.

2x — 5 = 20 + 5
Here, 2x — 5 is the left, and 20 + 5 is the right.

To move terms to the other side means to cancel out a term from one side. As a logical consequence of having to perform the same operation on both sides, the side that contained the term will cancel that part out, and the other side will have the term with a reversed sign.

If we want to move -5 to the right, we need to remove it from the left but add it to the right.
2x — 5 = 20 + 5 | +5 ⇒ 2x — 5 + 5 = 20 + 5 + 5 ⇒ 2x = 20 + 10 ⇒ 2x = 30

To summarize, to move a term to the other side, perform the opposite operation. For instance, if you have 2x — 4 = 10 and you want to move the 4, you add 4 to both sides.

Once again, the reverse operations are:

Addition/Subtraction (One the opposite of the other and vice-versa),
Multiplication/Division, and
Exponentiation/Logarithms/Radicals (More complex, usually not necessary).

Common Mistakes and Trying to Avoid Them

Mistakes happen often, especially when you’re learning. Here are the most common errors and tips to avoid them:

Forgetting to perform an operation on both sides of an equation.

For example, 2x = 10 cannot mean that x = 10, because we need to balance out the scales on both sides.
2x = 10 | / 2 ⇒ 2x / 2 = 10 / 2 ⇒ x = 5 is the correct way.

Forgetting that you need to switch the sign when moving a term from side to side.

In 2x + 2 = 10, we can’t move 2 without changing the sign, i.e. 2x + 2 = 10 ⇒ 2x ≠ 10 + 2
We need to change the sign: 2x + 2 = 10 | — 2 ⇒ 2x = 10–2

Some Problems to Try

Here are a few practice problems that could help you stick the concept of equations into your memory.

Easy

x + 3 = 7
8 = x — 4
3x = 9

Moderate

2x — 5 = 9
5x — 10 = 20
10 = 20 — x

Challenging

3x + 2 = x + 8
5x + 3 = 10 — 2x
2x = 4x — 20

Conclusion

I hope that this tutorial has helped you better grasp the concept of equations. As always, remember that practice makes perfect, so while you might not get it right away, you definitely can and will be able to after a bit of practice!

Thank you for reading, I look forward to any possible questions you might have — or answers to the practice problems — in the comments. Goodbye!