Equations Facts 1 • Astro Theme OpenBlog

Equations

An equation is a mathematical statement that shows the equality of two expressions.
It consists of two expressions separated by an equal sign.
The expressions on both sides of the equal sign are called the left-hand side (LHS) and the right-hand side (RHS) of the equation.
An equation can have variables, constants, coefficients, and mathematical operations such as addition, subtraction, multiplication, and division.
The solution of an equation is the value of the variable that satisfies the equation.
Equations can be linear or nonlinear based on the degree of the variable.
Linear equations have a degree of 1, while nonlinear equations have a degree greater than 1.
Equations can be solved using algebraic methods such as substitution, elimination, and graphical methods.
Equations are used in various fields of mathematics, science, engineering, and everyday life to model real-world problems.
Equations can be represented using mathematical notation, symbols, and formulas to express relationships between quantities.

// The below code will represent a linear equation.
2x + 3 = 7

// The below code will represent a system of linear equations.
2x + 3y = 7
3x - 2y = 4

// The below code will represent a quadratic equation.
x^2 + 2x + 1 = 0

// The below code will represent a system of nonlinear equations.
x^2 + y^2 = 25
x + y = 7

// The below code will represent a trigonometric equation.
sin(x) + cos(x) = 1

// The below code will represent a differential equation.
dy/dx = 2x

// The below code will represent a partial differential equation.
∂u/∂t = ∂^2u/∂x^2

// The below code will represent a matrix equation.
Ax = b

// The below code will represent a vector equation.
a . b = |a||b|cos(θ)

// The below code will represent a polynomial equation.
x^3 + 2x^2 - 3x + 1 = 0

// The below code will represent a simultaneous equation.
2x + 3y = 7
3x - 2y = 4

// The below code will represent a transcendental equation.
e^x = 2

// The below code will represent a parametric equation.
x = cos(t)
y = sin(t)

// The below code will represent a cubic equation.
x^3 + 2x^2 - 3x + 1 = 0

// The below code will represent a quartic equation.
x^4 + 2x^3 - 3x^2 + 4x - 1 = 0

// The below code will represent a quintic equation.
x^5 + 2x^4 - 3x^3 + 4x^2 - 5x + 1 = 0

Word Problems

Read the problem carefully and identify the unknown quantity or variable.
Assign a variable to represent the unknown quantity.
Translate the information given in the problem into an equation using the variable.
Solve the equation to find the value of the unknown quantity.
Check the solution by substituting the value back into the original problem.
Write the solution in the context of the problem.

Example:

Problem: The sum of two numbers is 15. If one number is 3 more than the other, find the numbers.

Solution:

Let the first number be x.
Let the second number be x + 3.

According to the problem, the sum of the two numbers is 15.
Therefore, x + (x + 3) = 15

Solving the equation, we get:
2x + 3 = 15
2x = 12
x = 6

The first number is 6.
The second number is 6 + 3 = 9.

Therefore, the two numbers are 6 and 9.

Types of Equations

Linear Equation: An equation of the form ax + b = 0, where a and b are constants and x is the variable, is called a linear equation.
Quadratic Equation: An equation of the form ax^2 + bx + c = 0, where a, b, and c are constants and x is the variable, is called a quadratic equation.
Cubic Equation: An equation of the form ax^3 + bx^2 + cx + d = 0, where a, b, c, and d are constants and x is the variable, is called a cubic equation.
Quartic Equation: An equation of the form ax^4 + bx^3 + cx^2 + dx + e = 0, where a, b, c, d, and e are constants and x is the variable, is called a quartic equation.
Quintic Equation: An equation of the form ax^5 + bx^4 + cx^3 + dx^2 + ex + f = 0, where a, b, c, d, e, and f are constants and x is the variable, is called a quintic equation.
Simultaneous Equations: A set of two or more equations that share the same variables is called a system of simultaneous equations.
Trigonometric Equations: Equations involving trigonometric functions such as sine, cosine, tangent, etc., are called trigonometric equations.
Exponential Equations: Equations involving exponential functions such as e^x, 2^x, etc., are called exponential equations.
Logarithmic Equations: Equations involving logarithmic functions such as log(x), ln(x), etc., are called logarithmic equations.
Polynomial Equations: Equations involving polynomial functions such as x^2 + 2x + 1, x^3 + 2x^2 - 3x + 1, etc., are called polynomial equations.
Rational Equations: Equations involving rational functions such as 1/x, x/(x+1), etc., are called rational equations.
Differential Equations: Equations involving derivatives of functions such as dy/dx, d^2y/dx^2, etc., are called differential equations.
Partial Differential Equations: Equations involving partial derivatives of functions such as ∂u/∂t, ∂^2u/∂x^2, etc., are called partial differential equations.
Matrix Equations: Equations involving matrices such as Ax = b, where A is a matrix, x is a vector, and b is a vector, are called matrix equations.
Vector Equations: Equations involving vectors such as a . b = |a||b|cos(θ), where a and b are vectors, are called vector equations.
Parametric Equations: Equations involving parameters such as x = cos(t), y = sin(t), etc., are called parametric equations.

Properties of Equations

Reflexive Property: An equation is always equal to itself. For example, a = a.
Symmetric Property: If two expressions are equal, then they can be interchanged without changing the equality. For example, if a = b, then b = a.

// Reflexive Property
a = a

// Symmetric Property
If a = b, then
b = a