In mathematics, an equation refers to a mathematical statement, which consists of an equal symbol (that is '=') between the two algebraic expressions that have the same value. The most common and basic algebraic equations in maths usually have one or more variables.
For instance, 2a + 3 = 20 is an equation in which 2a + 3 and 20 are two expressions separated by an ‘equal’ sign.
In an algebraic equation, the left-hand side is equal to the right-hand side.
For example,
8a + 6 = 13
Here, 8a + 6 is the expression on the left-hand side, which is equal to the expression 13 on the right-hand side.
Expression : 8a + 6 = 13
8 is Coefficient
a is variable
+ is operator
6,13 are constant
The process of finding the value of the variable is called solving the equation.
Example: 3x – y + 12, is not an equation, because it does not consist of an equals sign. It is only an expression.
Different Types of Equations:
Below are different types of equations, which we use in algebra to solve them.
Linear Equation
Quadratic Equation
Radical Equation
Exponential Equation
Rational Equation
Linear Equation:
A linear equation is an equation for a straight line.
Example: y = 2x + 1
5x – 1 = y
The term involved in the linear equation is either a constant or single variable or a product of a constant.
Linear equation will be written as:
Y = mx + c, m is not equals to 0.
where,
m is the slope
c is the point on which it cut y-axis
The following are examples of some linear equations:
with one variable: 5x - 10 = 2
with two variables: 5x + y = 3
Quadratic Equation:
A quadratic equation is a polynomial whose highest power is the square or variable (x2, y2, etc.)
The quadratic equation is a second-order equation in which any one of the variable contains an exponent of 2
The standard form of the quadratic equation is:
ax2 + bx + c = 0
Where, a, b, c are numbers
a, b are called the coefficients of x2 and x respectively, and c is called the constant.
The following are examples of some quadratic equations:
x2 + y + 7 = 0 where a=1, b= 1 and c = 7
2x2- 3y + 3 = 0 where a= 2, b= -3, c = 3
5x2 + 2y = 0 where a=5, b=2, c= -8
9x2 = 4
9x2 – 4 = 0 where a= 9, b=0 and c= -4
Radical Equation:
A radical equation is an equation in which a variable is under a radical.
Methods to solve the radical equations are:
Separate the radical expression involving the variable, in case of more than one radical expression, and then separate one of them.
Raise both sides to the index of the radical.
Example:
Solve \[\sqrt{5a^{2}+3a}\] - 2 = 0
Isolate the radical expression.
\[\sqrt{5a^{2}+3a}\] = 2
Raise both sides to the index of the radical; in this case, square both sides.
(\[\sqrt{5a^{2}+3a}\])2 = ( 2 )2
5a2 + 3a = 4
5a2 + 3a – 4 = 0Exponential Equation:
Exponential equations have variables in place of exponents. An exponential equation can be solved as:
ax = ay ; x = y
Example:
2x = 4
The above equation is equivalent to 2x = 22
Rational Equations:
A rational equation involves the rational expressions (in the form of fractions), with a variable, say x, in the numerator, denominator, or both.
Example : \[\frac{x}{2}\] = \[\frac{x+3}{4}\]
Let us solve the equation by cross multiplication and equating the like terms.
So, the rational equation becomes:
4x = 2(x + 3)
4x = 2x + 6
4x – 2x = 6
2x = 6
x = 6/2
x = 3
Examples:
Q. Solve the given equation:
2x – 6(2 - x) = 3x + 2
A: Simplify the given equation:
2x – 6(2 - x) = 3x + 2
2x – 12 + 6x = 3x + 2
8x = 3x + 14
8x - 3x = 14
5x = 14
x = 14 / 5
Therefore the solution for the given equation 2x – 6(2 - x) = 3x + 2 is 14/5.
Verification:
Substitute x = 14/5 in the given equation 2x – 6(2 - x) = 3x + 2 ;
2(14/5) – 6(2 - 14/5) = 3(14/5) + 2
28 / 5 + 24 / 5 = 52/5
28 + 24 = 52
52 = 52
L.H.S = R.H.S
Hence verified.
