An equation is a mathematical statement that includes the symbol 'equal to' between two algebraic expressions having equal values.
In the figure, 5x + 9 is the left-hand side expression corresponding to the right-hand side expression 24.
An equation is analogous to a weighing scale with equal weights on both sides.
It remains still as long as we add or remove the same amount from both sides of an equation. Likewise, if we multiply or divide the same number into both sides of an equation, the result remains the same.
For the above equation, 5x + 9 = 24,
We get 5x = 24 – 9 = 15.
Hence, we get x = 15 / 5 = 3.
We get the value of the variable as 3 after solving the equation.
In mathematics, there are three types of equations.
A linear equation with variables x and y has the following standard form:
Ax + By = C
A straight line is the graph of a linear equation with one or two variables.
A quadratic equation with variables x and y has the following standard form:
ax2 + bx + c = 0
We solve these kinds of equations by using the quadratic formula:
The quadratic equation's curve is shaped like a parabola.
A cubic equation with variables x and y has the following standard form:
ax3 + bx2 + cx + d = 0
The variable values that make an equation true is the solution or root of the equation.
If we add, subtract, multiply, or divide the same number on both sides of an equation, the result remains unchanged.
Equations are used to describe geometric shapes in Cartesian geometry. Because the equations under consideration, such as implicit equations or parametric equations, contain an unlimited number of solutions, so, instead of directly presenting the solutions or counting them, which is impossible, one uses equations to examine the characteristics of figures. This is the fundamental concept of algebraic geometry, a significant branch of mathematics.
Algebra is the study of two types of equations: polynomial equations and the specific case of linear equations.
Polynomial equations have the form:
P(x) = 0, where P is a polynomial,
While linear equations have the form ax + b = 0, where a and b are parameters with only one variable.
To solve equations from either family, we use algorithmic or geometric approaches derived from linear algebra or mathematical analysis. Algebra also investigates Diophantine equations with integer coefficients and solutions. The approaches used are unique and derived from number theory.
Differential equations involve one or more functions and their derivatives. We solve them by locating a function expression that does not use derivatives. Differential equations are used to represent processes with variable rates of change. We use them in fields of physics, chemistry, biology, and economics.