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Equation in Math - Definition, Types, and Parts

TL;DR

An equation in math is a statement that two expressions are equal, separated by an equals sign — for example, 3x + 5 = 14. This article covers the formal definition, every part of an equation, the seven main types you'll meet in school, and the most common mistakes when solving them.

What Is an Equation in Math?

An equation in math is a mathematical statement that two expressions have the same value, written with an equals sign (===) between them. The expression on the left is the left-hand side (LHS); the one on the right is the right-hand side (RHS).

$$\underbrace{3x + 5}{\text{LHS}} ;=; \underbrace{14}{\text{RHS}}$$

An equation is a claim — it says these two things are equal. Solving the equation means finding the value (or values) of the variable that make the claim true. In 3x + 5 = 14, the variable x must equal 3 — because 3(3) + 5 = 14. Any other value of x makes the claim false.

The defining feature of an equation is the equals sign. Drop it, and you have an expression, not an equation — and expressions can be simplified, but cannot be solved.

The Parts of an Equation

Every equation is built from the same small set of pieces. Once you can name each piece, every equation looks the same regardless of how complex it gets.

Take this equation:
5x² − 3x + 7 = 22

The parts are:

• Variable — the letter standing for an unknown number. Here, x.
• Constant — a fixed number with no variable attached. Here, the 7 on the left and 22 on the right are constants.
• Coefficient — the number multiplying a variable. In 5x², the coefficient is 5. In −3x, the coefficient is −3.
• Term — each separate piece added or subtracted. The LHS has three terms: 5x², −3x, and 7. The RHS has one term: 22.
• Operators — the symbols indicating operations: +, −, ×, ÷, exponents.
• Equals sign — the = separating the two sides. Without it, the line is an expression, not an equation.
• Left-hand side (LHS) — everything to the left of the equals sign. Here, 5x² − 3x + 7.
• Right-hand side (RHS) — everything to the right. Here, 22.
• Solution — the value(s) of the variable that make the LHS equal the RHS. Solving this equation requires moving everything to one side and applying the quadratic formula.

The Seven Main Types of Equations

Equations get classified by what kinds of operations and powers appear in them. Here are the seven types every algebra student should be able to recognize.

1. Linear Equation

A linear equation has the variable raised only to the first power. General form in one variable:

ax + b = 0, a ≠ 0

Example: 3x + 5 = 14. The graph is a straight line. Linear equations have exactly one solution.

2. Quadratic Equation

A quadratic equation has the variable raised to the second power as its highest term. General form:

ax² + bx + c = 0, a ≠ 0

Example: x² − 5x + 6 = 0. The graph is a parabola. Quadratic equations have up to two solutions.

3. Polynomial Equation

A polynomial equation has the variable raised to non-negative integer powers, with the highest power being any positive integer. General form (degree n):

an xⁿ + an−1 xⁿ⁻¹ + … + a1 x + a0 = 0

Linear and quadratic equations are special cases of polynomial equations (degree 1 and 2). Cubic (n = 3) and quartic (n = 4) equations also have closed-form solution formulas; degree-5 and higher generally do not.

4. Rational Equation

A rational equation has variables in the denominator of a fraction. Example:

1/x + 1/(x + 1) = 5/6

Solve by multiplying through by the common denominator. Watch for extraneous solutions — values that make a denominator zero must be excluded.

5. Radical Equation

A radical equation has the variable inside a square root, cube root, or higher root. Example:

√(x + 3) = 5

Solve by isolating the radical and then raising both sides to the appropriate power. Always check the answer — squaring both sides can introduce solutions that don't satisfy the original equation.

6. Exponential Equation

An exponential equation has the variable in the exponent. Example:

2^x = 3

Here, x = 5 because 2^5 = 32. For more complex cases, take the logarithm of both sides.

7. Logarithmic Equation

A logarithmic equation has the variable inside a logarithm. Example:

log₂(x) = 3

Here, x = 2³ = 8. Logarithmic and exponential equations are inverses of each other.

What Is the Difference Between an Equation and an Expression?

An equation has an equals sign. An expression doesn't. That's the whole distinction — but it has consequences students miss.

Worked comparison.

• Expression: 2x + 7. You can evaluate it at x = 3 to get 13. You can simplify if combined with another expression. You cannot "solve" it — there's nothing to solve for.
• Equation: 2x + 7 = 19. You can solve it: 2x = 12, x = 6. The equation makes a claim about x; the expression doesn't.

How Do You Solve an Equation?

Solving an equation means finding the value (or values) of the unknown that make the equation true. The general process boils down to four moves, repeated until the unknown sits alone on one side.

The four legal moves — each preserves equality:

1. Add the same quantity to both sides.
2. Subtract the same quantity from both sides.
3. Multiply both sides by the same non-zero quantity.
4. Divide both sides by the same non-zero quantity.

That's it. Every algebraic solving step is one of these four — or a combination.

Worked example. Solve 5x − 3 = 2x + 12.

1. Subtract 2x from both sides → 3x − 3 = 12.
2. Add 3 to both sides → 3x = 15.
3. Divide both sides by 3 → x = 5.
4. Check. 5(5) − 3 = 12 ✓.

The order rule. Work from the outside in — undo the operations in the reverse of the order they were applied.

Common Mistakes Solving Equations

Three failure modes dominate classroom errors when solving equations.

Mistake 1: Doing the same operation to only one side

Where it slips in: Multi-step solving where the student forgets that both sides must be treated identically.
Don't do this: 3x + 5 = 14, then writing 3x = 14 (only subtracting 5 from the LHS).
The correct way: Subtract 5 from both sides: 3x + 5 − 5 = 14 − 5, giving 3x = 9.

Mistake 2: Confusing equation with expression

Where it slips in: Treating 3x + 5 as something to "solve" when there's no equals sign.
Don't do this: Asked to simplify 3x + 5, writing x = −5/3. There's no equation here — only an expression.
The correct way: Expressions get simplified. If there's no equals sign, the answer is the simplified expression itself.

Mistake 3: Squaring or multiplying out without checking

Where it slips in: Radical and rational equations, where squaring both sides can introduce false solutions.
Don't do this: x = −3, then squaring to get x = 9.
The correct way: After squaring, substitute every candidate back into the original equation and discard any that don't check.

The Mathematicians Who Shaped the Equation

Diophantus of Alexandria (c. 200–284 CE, Egypt/Greece) — The "father of algebra" in the Greek tradition.
Muhammad ibn Musa al-Khwarizmi (c. 780–850, Persia/Baghdad) — Gave the world the first systematic procedures for solving linear and quadratic equations.
Robert Recorde (1510–1558, Wales) — Invented the equals sign in 1557.
René Descartes (1596–1650, France) — Standardised the modern notation for equations.

A Practical Next Step

Try these three problems before going deeper into specific equation types.

1. Solve 4x − 7 = 13. (Linear; one step at a time.)
2. Solve x² − 7x + 12 = 0 by factoring. (Quadratic; find two numbers that multiply to 12 and add to −7.)
3. Solve √(x + 4) = 3. (Radical; isolate the root, then square — and check.)

Frequently Asked Questions

What is an equation in math?
An equation is a mathematical statement that two expressions are equal, written with an equals sign between them.

What is the difference between an equation and an expression?
An equation has an equals sign and can be solved; an expression doesn't and can only be simplified.

What are the parts of an equation?
The parts are: variables, constants, coefficients, terms, operators, the equals sign, and the solution.

What are the seven main types of equations?
Linear, quadratic, polynomial, rational, radical, exponential, and logarithmic.

Who invented the equals sign?
Robert Recorde in 1557.

How do you solve an equation?
Isolate the variable while keeping the equation balanced.

How many solutions does an equation have?
It depends on the type of equation.