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# Equation|Definition & Meaning

**Definition**

**Equations** are **mathematical assertions** that relate **expressions** with one another using the **equality** symbol. In other words, it equates two things, meaning they are the same. Using equations, we can find the **exact relationship** between the mathematical expressions involved and even **variable values**.

It illustrates the** mathematical equivalence** of **one expression** with another expression and **helps** in **finding the value** of an **undetermined variable.**

**Illustration of Equation **

Figure 1 – Illustration of Equation

Suppose an equation as a** weigh beam** or** scale**, we know that the** weigh beam** **has two sides** and there is a **center in between** these** sides**, **treat** the **center as** an **equality sign** and **suppose** there are **a and b quantities** on the** left side** and **right side** and **c quantity** on the **right side.** **Consider** them as **weight**, in order **to have** the equivalency between the **left and ****right portions as shown **in the figure we **have to make** a **balance between** the **two sides** by **either adding weight** or **removing weight**.

**When** **we** are **able to** **balance** the** two sides** we can take this scenario as an** analogy to the equation** where the **left side represents** **one mathematical expression** and the **right side represent** **another expression** while the **center** of the **weighing beam** is **equality** which gives the **mathematical equivalence** between two sides. This concept is illustrated in the figure above.

**Components of Equations**

Figure 2 – Component of equation

There are **various components** that makeup and equations.

**Coefficient**

**Mathematicians use** coefficients **to multiply terms** in polynomials, series, and expressions; they are **usually numbers**. It is also known as the **factor of multiplication.**

**Variable**

**Symbols and placeholders** are **used to represent** **mathematical objects** in mathematics. There are **various types** of **variables**, including **numbers, vectors, matrices, functions, arguments, sets, and elements.**

**Operator**

Mathematical **operators** are** symbols** that **indicate how an operation** will be** carried out**. For instance **addition, subtraction, division, and multiplication.**

**Constant**

Typically, **mathematicians refer** to **mathematical constants** by their **names or symbols** (e.g., alphabet letters) that are **unambiguous** in their meaning, enabling them to be **used across** a **variety of mathematical** **situations** without changing their values.

**Expression**

An expression is the** amalgam** of **coefficient, variable, constant and mathematical operator without** an **equality sign**.

All the components of the equations are shown in the figure above.

**Categories of Equations**

Although there are **many types of equations, **we generally divide them into **three categories**:

- Linear Equations
- Quadratic Equations
- Cubic Equations

**Linear Equation**

**An equation of first order is a linear equation.** A linear equation in a coordinate system is defined by lines. We can define a **linear equation** in **one variable** as the **equation containing** only one **homogeneous variable of degree 1**.

There is **no restriction on either using one variable or many **in a linear equation. It is called **linear equations in two variables**, and **so on**, **if** there are **two variables** in the l**inear equation**. For instance, 3y = 2x + 4 and 5x – 3y + 9 = 0 are linear equations.

**Quadratic Equation**

**Approximately** a** quadratic equation** is a **polynomial equation** that has a **degree of 2**, such as f(x) = Lx^{2} + Mx + N = 0 where** L, M, and N** belong to **real numbers**. The so-called **leading coefficient** L is used to **denote** the** leading term** of f (x), and the **absolute term N** is used to **denote** the** absolute term**.

L** is** defined as a **positive quadratic equation** when its **roots satisfy its equation (α, β)**. Two roots are always present in the quadratic equation. In addition to **real roots**, **imaginary roots** can also exist. **Adding a zero** to **quadratic polynomial** **results** in a **quadratic equation**. The **values of L** that **satisfy the quadratic equation are roots.**

**Cubic Equation**

**Mathematicians** **express cubic equations** with the **cubic equation formula**. The term cubic equation **refers to an equation** with **degree three.** The** roots of cubic equations** are **1 real and 2 imaginary or 3 real roots**. In mathematics, **cubic polynomials** are **polynomials with three degrees**. The cubic equation is written as

ax^{3} + bx^{2} + cx = d = 0

**Comparison of Equation With Expression**

**Properties of Equations**

- An
**equation**is**represented**by**an equality sign**placed between**two mathematical expressions**to**show**the**mathematical equivalence**. - The
**equal sign**is**mandatory**for equation representation. **Undetermined values**can be**computed**with**equations**. For instance, 2x + 3x + 2 = 0 has the solution x = -2/5.

**Properties of Expression**

- Unlike equations,
**expression is represented**by just a**mathematical expression without**the**“=” sign**. - There is
**no equality sign**in the expression, it**contains only**the**variable, coefficient, mathematical operator, and constant.** - The
**expression**can only be**simplified further**and**doesn’t provide knowledg**e to**determine unknown quantities.**For instance, 2x + 4; we cannot compute x.

**Illustration of Important Equations**

**Equation of a Simple Line**

Figure 3 – Equation of line

The** equation** of a simple **line** **ax + by + c = 0** can be represented as **slope intercept form** as:

y = mx + c

Where **m** is the **slope**.

**Equation of X-axis**

Figure 4 – Equation of the x-axis.

For the **x-axis equation**, we will use the **standard form** of the **equation of the line** which is y = mx + c. We know that, at the** x-axis**, **y = 0. **Putting **y = 0** in the **above equation** **gives** the **equation of the x-axis** as:

x = c or y = 0

**Equation of Y-axis**

Figure 5 – Equation of y-axis.

For the **y-axis equation**, we will **use the standard form** **of** the **equation of the line** which is y = mx + c. We know **at the y-axis** **x=0** so putting **x=0** in the **above equation** **gives** the **equation of the x-axis** as:

y = c or x = 0

**Examples of Equations**

**Example 1**

Consider **two equations: **2x+4y=2 and 2x+3y=1. **Compute** the **unknown value of x and y**.

**Solution**

2x + 4y = 2 [Eq. 1]

2x + 3y = 1 [Eq. 2]

**Subtracting equations 1 and 2** gives:

2x + 4y = 2

-2x – 3y = 1

y = 1

**Put the value of y** in equation 1:

2x + 4(1) = 2

2x + 4 = 2

2x = 2 – 4

2x = -2

x = $\mathsf{\dfrac{-2}{2}}$

So, x=-1 and y=1.

**Example 2**

Consider** two equations** x^{2}-16=0 and x^{2}+6x=0. **Solve** for the **value of x**.

**Solution**

**Equation 1**

x^{2 }– 16 = 0

x^{2} = 16

**Taking square root** on both sides:

\[ \mathsf{\sqrt{x^2} = \sqrt{16}} \]

x = $\pm$4

**Verify** by putting the value of x in the original equation:

x^{2 }– 16 = 0

4^{2 }– 16 = 0

16 – 16 = 0

**Equation 2**

x^{2 }+ 6x = 0

Taking **x** as **common:**

x(x + 6) = 0

It implies that:

x = 0

x = -6

**Verify** by putting the value of x in the original equation:

(-6)^{2} + 6(-6) = 0

36 – 36 = 0

*All mathematical drawings and images were created with GeoGebra.*