Equations: A Basic Introduction
Math teachers love to say “let x equal the unknown.” But nobody explains why that simple trick is so powerful. Here’s the thing: equations are the reason we can solve problems that would take hours of guesswork in seconds. Let me show you how.
Why Equations Matter
Applied mathematics isn’t abstract theory—it’s the toolkit you use to solve real problems. Business decisions, engineering calculations, financial planning—they all boil down to equations.
But before we dive into the theory, let’s see equations in action with three progressively harder examples.
Example 1: George’s Money
Problem: George had some money. He gave $14 to Matthew. Now he has $27. How much did George start with?
You probably already know the answer: $41. George had $41, gave away $14, and was left with $27. Simple mental math.
But here’s what makes equations powerful: they let you solve problems where mental math fails. Let me show you the mathematical approach.
Step 1: Translate words into symbols
Replace “some money” with a variable. I’ll use \( x \).
| What Happened | Mathematical Expression |
|---|---|
| George started with | \( x \) dollars |
| He gave Matthew | 14 dollars |
| He now has | \( x – 14 \) dollars |
The problem tells us he has $27 left. So:
\( x – 14 = 27 \)
That’s an equation. It’s a statement containing an unknown quantity and an equals sign.
Step 2: Solve for x
Add 14 to both sides:
$$ x – 14 + 14 = 27 + 14 $$
$$ x = 41 $$
George had $41. Same answer as our mental math—but now we have a method that scales to harder problems.
Pro Tip: When you move a term across the equals sign, flip its sign. Here, \( -14 \) becomes \( +14 \). This is just a shortcut for “add 14 to both sides.”
What Exactly Is an Equation?
An equation is a mathematical statement that says two things are equal. It contains:
- Variables — Unknown quantities, usually represented by letters from the end of the alphabet: \( x, y, z, \omega, \theta \)
- Constants — Known values, often represented by letters from the beginning of the alphabet: \( a, b, c, d \)
- An equals sign — The bridge that makes it an equation, not just an expression
Solving an equation means finding the value(s) of the variable that make the statement true.
The Four Operations You Need
To solve any basic equation, you only need four operations and one rule:
| Operation | What It Does | When Moving Across = |
|---|---|---|
| Addition (+) | Increases a value | Becomes subtraction |
| Subtraction (−) | Decreases a value | Becomes addition |
| Multiplication (×) | Scales a value | Becomes division |
| Division (÷) | Splits a value | Becomes multiplication |
The Golden Rule: Whatever you do to one side of an equation, you must do to the other side. That’s it. That’s the whole secret.
Example 2: Mary’s Sheep
Problem: Mary had 7 sheep. Her uncle gifted her some more. She now has 18 sheep. How many did her uncle give her?
Let \( x \) = the number of sheep from her uncle.
\( 7 + x = 18 \)
Move 7 to the other side (it becomes −7):
\( x = 18 – 7 = 9 \)
Answer: Mary’s uncle gifted her 9 sheep.
Note: Numbers without a sign in front are positive. So \( 7 \) and \( +7 \) mean the same thing.
Example 3: The Cricket Ball Problem
Problem: Monty has some cricket balls. Graham has twice as many as Monty. Adam has 6 balls. Together, they have 27 balls. How many does each person have?
This is where equations really shine. Mental math gets messy here, but algebra keeps things clean.
Step 1: Define variables
- Monty has \( x \) balls
- Graham has \( 2x \) balls (twice Monty’s amount)
- Adam has 6 balls
Step 2: Write the equation
\( x + 2x + 6 = 27 \)
Step 3: Simplify and solve
\( 3x + 6 = 27 \)
\( 3x = 27 – 6 = 21 \)
\( x = \frac{21}{3} = 7 \)
Answer: Monty has 7 balls, Graham has 14 balls, Adam has 6 balls. Total: \( 7 + 14 + 6 = 27 \). ✓
Notice how the multiplication by 3 became division when we moved it across the equals sign. Same principle—inverse operations.
Types of Equations: A Complete Classification
Algebraic equations (we call them “algebraic” because they contain variables, which belong to algebra) are classified in two main ways:
- By number of variables — How many unknowns does the equation contain?
- By degree — What’s the highest power of the variable?
Classification by Number of Variables
| Type | Variables | Examples |
|---|---|---|
| Univariable | 1 | \( 3x + 2 = 11 \), \( x^2 – 4 = 0 \) |
| Bivariable | 2 | \( x + y = 5 \), \( x^2 + y^2 = 25 \) |
| Trivariable | 3 | \( x + y + z = 10 \), \( x^2 + y^2 – z^2 = 0 \) |
| n-variable | n | Equations with n different unknowns |
Classification by Degree (For Univariable Equations)
The degree of an equation is the highest power of the variable. This determines how many solutions the equation can have and what methods you need to solve it.
| Name | Degree | General Form | Max Solutions |
|---|---|---|---|
| Linear | 1 | \( ax + b = 0 \) | 1 |
| Quadratic | 2 | \( ax^2 + bx + c = 0 \) | 2 |
| Cubic | 3 | \( ax^3 + bx^2 + cx + d = 0 \) | 3 |
| Quartic | 4 | \( ax^4 + bx^3 + cx^2 + dx + e = 0 \) | 4 |
| n-th degree | n | Highest power is \( x^n \) | n |
Quick Reference: In these general forms, \( a, b, c, d, e \) are constants (known values), and \( x \) is the variable (what we’re solving for). The coefficient \( a \) cannot be zero—otherwise, the equation drops to a lower degree.
Examples of Each Degree
- Linear: \( 3x + 2 = 5x – 3 \) — Variable appears only to the first power
- Quadratic: \( x^2 + 5x + 3 = 0 \) — Highest power is 2 (also called “square equations”)
- Cubic: \( x^3 – 6x^2 + 11x – 6 = 0 \) — Highest power is 3
- Special: \( e^x = x^e \) — Here, \( e \) is Euler’s number (a constant ≈ 2.718), and \( x \) is the variable
Bivariable Equations in Detail
Equations with two variables follow similar classification rules, but the “degree” works slightly differently:
- Linear bivariable: The power of each variable (or sum of powers in a product term) doesn’t exceed 1
- \( ax + by = c \) is linear ✓
- \( axy = b \) is NOT linear (the \( xy \) term has combined degree 2) ✗
- Second-order bivariable: Maximum combined degree is 2
- \( ax^2 + by^2 + cxy + dx + ey + f = 0 \)
- This includes circles, ellipses, parabolas, and hyperbolas
Frequently Asked Questions
What’s the difference between an expression and an equation?
An expression is a mathematical phrase like 3x + 5 — it doesn’t equal anything. An equation has an equals sign connecting two expressions, like 3x + 5 = 20. You can simplify expressions, but you solve equations.
Why do we use letters like x and y instead of just question marks?
Letters let us manipulate unknowns using the same rules as numbers. You can add x to both sides of an equation, multiply 3 by x to get 3x, or substitute a value for x. Try doing that with a question mark! The convention of using x, y, z for variables and a, b, c for constants comes from René Descartes in the 1600s.
Can an equation have no solution or infinitely many solutions?
Yes to both! The equation x + 1 = x + 2 has no solution (nothing makes 1 = 2 true). The equation 2(x + 1) = 2x + 2 has infinitely many solutions — every value of x works because both sides are always equal. These are called contradictions and identities, respectively.
What does it mean to ‘solve’ an equation?
Solving an equation means finding all values of the variable that make the equation true. For 2x = 10, the solution is x = 5 because substituting 5 gives 2(5) = 10, which is true. The solution is also called the ‘root’ of the equation.
Why does moving a term across the equals sign flip its sign?
It’s a shortcut for doing the same operation to both sides. When you ‘move’ +5 from the left to the right and it becomes -5, you’re really subtracting 5 from both sides. The shortcut just skips writing the intermediate step. Same principle: multiplication becomes division, and division becomes multiplication.
Key Takeaways
- Equations translate word problems into solvable math — Replace “some unknown” with a variable
- The golden rule: Whatever you do to one side, do to the other
- Moving terms flips operations: + becomes −, × becomes ÷
- Classification matters: The type of equation determines which solving methods work
- Degree = maximum power: Linear (1), Quadratic (2), Cubic (3), and so on
Ready to go deeper? Next up: Solving Linear Equations — From Basics to Advanced Techniques
Disclaimer: My content is reader-supported, meaning that if you click on some of the links in my posts and make a purchase, I may earn a small commission at no extra cost to you. These affiliate links help me keep the content on gauravtiwari.org free and full of valuable insights. I only recommend products and services that I trust and believe will genuinely benefit you. Your support through these links is greatly appreciated—it helps me continue to create helpful content and resources for you. Thank you! ~ Gaurav Tiwari