What Are Maths Equations Definition Types and How to Solve
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.
FAQs on Maths Equations Explained with Concepts and Methods
1. What is an equation in maths?
An equation is a mathematical statement that shows two expressions are equal using an equals sign (=). It typically contains numbers, variables, and mathematical operations.
- Example: 2x + 3 = 7
- The left-hand side (LHS) equals the right-hand side (RHS).
- Solving an equation means finding the value of the variable that makes the statement true.
2. What is a linear equation?
A linear equation is an equation where the highest power of the variable is 1. It can be written in the standard form ax + b = 0, where a ≠ 0.
- Example: 3x + 5 = 11
- Solving gives: 3x = 6 → x = 2
- Its graph is a straight line.
3. How do you solve a simple equation step by step?
To solve a simple equation, isolate the variable by performing inverse operations on both sides. Follow these steps:
- Step 1: Write the equation (e.g., 2x + 4 = 10).
- Step 2: Subtract 4 from both sides → 2x = 6.
- Step 3: Divide both sides by 2 → x = 3.
4. What is the quadratic equation formula?
The quadratic formula is x = (-b ± √(b² − 4ac)) / 2a and is used to solve equations of the form ax² + bx + c = 0.
- Example: x² − 5x + 6 = 0
- a = 1, b = −5, c = 6
- Solutions: x = 2 and x = 3
5. What is the difference between an equation and an expression?
An equation contains an equals sign (=) while an expression does not. An expression represents a value, whereas an equation states that two expressions are equal.
- Expression: 3x + 2
- Equation: 3x + 2 = 11
6. What are the types of equations in maths?
The main types of equations in maths include linear, quadratic, cubic, and simultaneous equations. Common types are:
- Linear equations (degree 1)
- Quadratic equations (degree 2)
- Cubic equations (degree 3)
- Simultaneous equations (two or more equations solved together)
7. How do you solve simultaneous equations?
Simultaneous equations are solved by finding values that satisfy both equations at the same time. One common method is substitution.
- Example: x + y = 5 and x − y = 1
- Add both equations → 2x = 6 → x = 3
- Substitute into x + y = 5 → y = 2
8. What is a balanced equation in maths?
A balanced equation means both sides have equal values and remain equal when the same operation is applied to both sides. This follows the equality property.
- If 4x = 12, dividing both sides by 4 keeps it balanced.
- Result: x = 3
9. What does it mean to check the solution of an equation?
Checking a solution means substituting the value back into the original equation to verify it makes the equation true.
- Example: Solve 2x = 8 → x = 4
- Substitute: 2(4) = 8
- Since 8 = 8, the solution is correct.
10. What are common mistakes when solving equations?
Common mistakes when solving equations include incorrect sign changes, not applying operations to both sides, and calculation errors.
- Forgetting to change signs when moving terms
- Dividing only one side instead of both
- Miscalculating squares or square roots in quadratic equations
