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## What Is the Difference Between Equation vs Expression

The main difference between an **expression** and an **equation** is that an **expression** represents a mathematical “phrase” comprising numbers, variables, and operations without an equal sign, while an **equation** is a mathematical “sentence” asserting the equality of two **expressions**, always including an equal sign.

Both **“equation”** and **“expression”** are fundamental terms in mathematics, and while they might sound similar, they refer to different concepts. Here’s a breakdown of their differences:

### Expression

An **expression** is a combination of numbers, variables, and operations. For example, **5 * x + 3** or **2 *$x^2$ – 7 * y + 10**. An **expression** does not have an equals sign (=). It doesn’t necessarily represent a complete thought in mathematical terms. Think of it as a phrase in the language of mathematics. **Expressions** have value. For instance, if you plug in a value for x in the **expression ****5 * x + 3**, you can evaluate the **expression** to get a single number.

Figure-1.

### Equation

An **equation** states that two **expressions** are equal, meaning they have the same value. For example, 2 + 3 = 5 or 5x + 3 = 23. An **equation** always includes an equals sign (=). It represents a complete thought, stating that two things are equal. Using the language analogy, it’s a complete sentence. **Equations** can be true or false. For example, the **equation** 3 + 2 = 6 is false, while the **equation** 3 + 2 = 5 is true. In summary, while both **expressions** and **equations** deal with numbers, variables, and operations, **expressions** are mathematical “phrases” that can be evaluated to give a value, whereas **equations** are mathematical “sentences” that state the equality between two **expressions**.

## Properties

Both **equations** and **expressions** have properties that derive from the foundational principles of algebra. Let’s delve into these properties:

### Expression Properties

#### Commutative Property (of Addition and Multiplication)

For any numbers a and b, the order in which you add or multiply them doesn’t change the result. a + b = b + a a × b = b × a

#### Associative Property (of Addition and Multiplication)

For any numbers a, b, and c, the way in which you group them when adding or multiplying doesn’t change the result. (a + b) + c = a + (b + c) (a × b) × c = a × (b × c)

#### Distributive Property

Multiplication distributes over addition. a × (b + c) = a × b + a × c

#### Identity Property

For addition, the identity is 0 because any number plus 0 is the number itself. For multiplication, the identity is 1 because any number times 1 is the number itself. a + 0 = a a × 1 = a

#### Inverse Property

For every number, there’s an additive inverse so that their sum is zero. Similarly, for every non-zero number, there’s a multiplicative inverse so that their product is one. a + (-a) = 0 a × (1/a) = 1 for a ≠ 0

### Equation Properties

#### Reflexive Property

Any quantity is equal to itself. a = a

#### Symmetric Property

If one quantity equals another, then the second is equal to the first. If a = b, then b = a

#### Transitive Property

If one quantity equals a second, and the second equals a third, then the first quantity equals the third. If a = b and b = c, then a = c

#### Substitution Property

If two quantities are equal, one can be substituted for the other in any **equation** or **expression**. If a = b, then a can replace b in any **expression** or **equation**.

#### Addition and Multiplication Properties of Equality

If you add or multiply each side of an **equation** by the same number, the two sides remain equal. If a = b, then a + c = b + c If a = b, then a × c = b × c

#### Division Property of Equality

If you divide each side of an **equation** by the same non-zero number, the two sides remain equal. If a = b and c ≠ 0, then a/c = b/c Remember that while properties of **expressions** are related to manipulating and simplifying them, properties of **equations** relate to maintaining equality and deducing relationships between quantities.

**Exercise **

**Example 1**

**Expression:** $4x+7$

Figure-2.

**Solution**

This is a simple algebraic **expression** in terms of $x$. Without a specific value for $x$, we cannot evaluate it further.

**Example 2**

**Expression:** $3$(y^2)$ −5y+8$

Figure-3.

**Solution**

This is a quadratic** expression** in terms of $y$. Its value depends on the value assigned to $y$. For example, if $y=2$, the value is:

3$(2^2)$$=10$.

**Example 3**

**Equation:** $2x+5=11$

**Solution**

To solve for $x$:

2x = 11−5

$2x=$6

$x=3$

**Example 4**

**Equation **$(y^2)$ $−4y+4=0$

**Solution**

This is a quadratic **equation**. Factorizing, $(y−2)(y−2)=0$ So, $y=2$ is the solution.

**Applications **

**Equations** and **expressions** are foundational concepts in mathematics and thus find applications across numerous fields. Here’s how they’re used in various domains:

**Pure Mathematics****Expression:**Mathematicians use**expressions**to define functions, such as $f(x)= $(x^2)$$ $+3x−7$. These functions can then be studied for their properties, like continuity or differentiability.**Equation:****Equations**like $(x^2)$$−5x+6=0$**(quadratic equations)**are solved to find values of $x$. In algebra,**equations**are used to prove theorems and solve for unknowns.

**Physics****Expression:**Physical quantities, like kinetic energy (1/2 $ m$(v^2)$$) or gravitational potential energy ($mgh$), are represented as**expressions**.**Equation:****Equations**describe laws of physics, such as Newton’s second law ($F=m * a$) or Einstein’s mass-energy equivalence ($E=m$c^_{2$}$).

**Engineering****Expression:**Engineers use**expressions**to represent quantities like stress, strain, and electrical resistance in a system.**Equation:****Equations**are essential in design calculations. For example, in electrical engineering, Ohm’s law ($V=IR$) is an**equation**used to relate voltage, current, and resistance.

**Economics****Expression:****Expressions**might represent economic variables like the utility function or production function.**Equation:****Equations**, like the equilibrium condition where supply equals demand, are pivotal in analyzing economic models and systems.

**Computer Science****Expression:**In programming,**expressions**evaluate a value, such as arithmetic computations or string manipulations.**Equation:**In some functional programming languages,**equations**define relationships between variables or functions. In algorithms,**equations**might describe relationships between variables or components.

**Biology and Medicine****Expression:****Expressions**might represent the rate of enzyme activity or the concentration of a solute in a solution.**Equation:****Equations**can model biological systems, like the Michaelis-Menten**equation**in enzyme kinetics or the exponential growth of bacterial populations.

**Finance****Expression:**Financial metrics, like earnings before interest and taxes (EBIT) or net present value (NPV), can be represented as**expressions**.**Equation:**In finance,**equations**are used to establish relationships, such as the one between different financial ratios or between present value and future value.

In all these fields and many more, **expressions** and **equations** allow professionals to represent, analyze, and predict various phenomena. The precise nature of their use depends on the specific requirements of each domain.

*All images were created with GeoGebra.*