Equation

Introduction to Equation

An equation is a mathematical sentence that shows that two sides are equal. There is always an equal sign (=) in the middle. For example, 3 + 2 = 5 is an equation because both sides represent the same value. An equation can only contain a mix of numbers and letters. The letters are called variables, and they stand for unknown numbers.

The main goal of solving an equation is to find the value of the variable. For example, in x + 4 = 9, we must find the number that makes this statement correct. If we think carefully, x = 5 because 5 + 4 = 9. When we solve the equations, we must remember that no matter what we do on one side, we should also do on the other side to keep both sides equal.

The equations are very useful in real life. They help us solve puzzles, calculate money, measure distance, and even solve science problems. The learning equations will make it easier to solve major problems in higher classes after the step-by-step guide. With practice, the equations become simple and fun to solve.

Table of Contents  

• Equation meaning

• Equation in algebra

• Types of Equations

• Equation of a Line 

• Equation of the X-axis 

• Equation in One Variable

• Solved Examples 

• Real-Life Applications of Equations

• FAQs on Equation

Equation meaning

In mathematics, an equation is a statement that shows that two things are equal. There is always an equal sign (=) in the middle. For example: 3x + 2 = 11

Here is the left side, 3x + 2, and the right side is 11. The equal sign "=" shows that both sides have the same value.

Parts of an equation

Let's look at parts of the equation 3x + 2 = 11:

• Left Hand Side (LHS): Part of the left side of the equal sign (3x + 2)

• Right Hand Side (RHS): Part of the right side of the equal sign (11)

• Equal sign (=): It connects both sides and shows that they are equal.

• So an equation combines two expressions and tells us that their values ​​are the same.

Understanding Equations with a Simple Example

Before we learn the definition of an equation, let's first see how an equation is formed with the help of a small example. Imagine we make squares using matchsticks.

• To make 1 square, we need 4 matchsticks.

• To make 2 squares in a row, we need 7 matchsticks.

• To make 3 squares in a row, we need 10 matchsticks.

• To make 4 squares in a row, we need 13 matchsticks.

We can put it in a table:

From the pattern, we can see that each new square adds 3 more sticks. So here's the rule:

• Number of matchsticks = 3x + 1, where x = number of squares.

• Suppose we want to find out how many squares can be formed with 40 matchsticks.

• According to our rules:

• 3x + 1 = 40

• 3x = 39

• X = 39 ÷ 3

• x = 13

• That's why we can make a 13-square with 40 matchsticks.
  
  

Equation in algebra

In algebra, an equation is a rule that includes a variable (a letter such as x or y). The equation is only true for a certain value of the variable. For example, the equation 3x + 4 = 10 is only true when x = 2 because 3 (2) + 4 = 6 + 4 = 10.

The difference between an expression and an equation is simple:

Types of Equations

In algebra, there are different types of equations. Some important ones are:

• Linear equations: Equations where the variable has a power of 1. Example: 2x + 5 = 9.

• Quadratic equations: Equations where the highest power of the variables is 2. Examples: bp x² + 3x + 2 = 0.

• Cubic equations: Equations where the highest power of the variable is 3. Example: x³ – 2x = 5.

• Quartic equations: Equations where the highest power of the variable is 4. Example: x⁴ – 3x² + 2 = 0.

• Differential equations: Equations that include the rate of change.

• Parametric equations: Equations written using another variable.
  
  

Equation of a Line 

A line in mathematics is straight and goes in both directions without ending. We can write a line equation in different ways. The most commonly used slope-intercept form:

y = mx + b

Here:

• m = slope of the line (it tells us how steep or slanted the line is).

• b = y-intercept (the point where the line touches the y-axis).

Example: If the equation is y = 2x + 3, the slope (m) is 2 and the line crosses the y-axis at the point (0, 3).

• Another way to write a line is the general form:

• Ax + By + C = 0

• Where A, B, and C are numbers. This is just another way to show the same line.
  
  

Equation of the X-axis 

The x-axis in the coordinate plane has a horizontal line (a flat line that goes to the left and right). On this line, the value of y is always 0, no matter what the value of x is. This means that the equation of the X-axis is y = 0.

For example:

• Point (2, 0) lies on the x-axis.

• The point (-5, 0) also lies on the x-axis.

• Even point (0, 0), which is the origin, is part of the x-axis.

• So any point on the x-axis can be written as (C, 0), where C can be any number (positive, negative, or zero).

• This makes the equation of the x-axis very simple: y = 0.
  
  

Equation in One Variable

An equation in one variable is an equation that has only one unknown letter. This variable is the number we need to find to make the equation true.

Examples:

Key points to remember:

• One variable equation: Only one unknown letter (like x).

• Multivariable equation: Two or more unknowns (like x, y, z)

• No variable:Just numbers, no unknown.
  
  

Solved Examples 

Example 1:

Use the linear formula:

Example: 2x + 3 = 7  

Add 3 to both sides: 2x = 10.

Divide by 2: x = 5.

Example 2:

Use the quadratic formula:

x = (b ± √(b² - 4ac)) / 2a  

Example: x² + 3x - 4 = 0.

x = (3 ± √(9 + 16)) / 2 = (3 ± 5) / 2.

x = -1 or x = 4.

Example 3:

Use logarithms or make the base the same.

Example: 2ˣ = 16 → 2ˣ = 2⁴ → x = 4.

Each of these processes helps simplify equations and find the correct solutions.

Example 4:

Solve the Linear Equation  

3x + 2 = 11  

Ans: 3x = 9 → x = 3.

Example 5:

Solve the Exponential Equation  

5ˣ = 125  

Ans: 5ˣ = 5³ → x = 3.

Example 6:

Solve the Rational Equation  

1/x + 1/2 = 3/4  

Ans: Solve using a common denominator → x = 4.

Example 7:

Solve the Radical Equation  

√(x + 3) = 5  

Ans: Square both sides → x + 3 = 25 → x = 22.

Real-Life Applications of Equations

Equations aren’t just in books; they are in everyday life:

•  Economics: Predicting profit and loss using linear equations.

•  Physics: Describing motion and forces using math equations.

•  Biology: Modelling population growth with exponential equations.

•  Engineering: Designing circuits and systems using quadratic equations.

•  Finance: Calculating interest with exponential equations.

Equations allow us to model and solve complex real-world problems.

FAQs on Equation

1. What are the 7 types of equations? 

Answer: The 7 types of equations are:

• Linear equations

• Quadratic equations

• Cubic equations

• Polynomial equations

• Rational equations

• Radical equations

• Exponential and logarithmic equations

2. What is a definitional equation in economics?

Answer: A definitional equation in economics is an identity used to define a concept, like GDP = C + I + G + (X - M), where each term defines a component of Gross Domestic Product.

3. What are some of the 7 hardest math equations?

Answer: The 7 hardest math equations (Millennium Prize Problems) are:

• Riemann Hypothesis

• Birch and Swinnerton-Dyer Conjecture

• Hodge Conjecture

• Navier-Stokes Equation

• Yang-Millss Existence and Mass Gap

• P vs NP Problem

• Poincaré Conjecture 

4. Can you give 4 simple examples of equations?

Answer: Four examples of equations are:

• Linear: y = 2x + 5

• Quadratic: x² + 3x + 2 = 0

• Logarithmic: log(x) = 3

• Rational: (x + 2)/(x -- 1) = 4