Step-by-Step Guide to Solving Maths Equations Easily
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 Essential Maths Equations Explained for Students
1. What is a mathematical equation and what are its main components?
A mathematical equation is a statement that asserts the equality of two mathematical expressions. It is distinguished by the presence of an equals sign (=). The main components of an equation are:
- Variables: Symbols (like x, y) that represent unknown quantities.
- Constants: Fixed numerical values.
- Coefficients: Numbers multiplied by variables.
- Operators: Mathematical symbols like addition (+), subtraction (-), multiplication (*), and division (/).
- Expressions: The combinations of variables, constants, and operators on both the Left-Hand Side (LHS) and Right-Hand Side (RHS) of the equals sign.
2. What are the main types of equations taught in the CBSE Maths syllabus?
The CBSE syllabus introduces different types of equations across various classes. The primary types include:
- Linear Equations: Equations of degree one. They can be in one variable (e.g., ax + b = 0) or two variables (e.g., ax + by + c = 0).
- Quadratic Equations: Polynomial equations of degree two, in the standard form ax² + bx + c = 0.
- Polynomial Equations: Equations with a degree higher than two, such as cubic (degree 3) and biquadratic (degree 4) equations.
- Trigonometric Equations: These equations involve trigonometric functions (sin, cos, tan, etc.) of a variable, for example, sin(x) + cos(x) = 1.
- Differential Equations: Advanced equations that involve an unknown function and its derivatives, such as dy/dx + y = 0.
3. Can you give some simple examples for different types of mathematical equations?
Certainly. Here are some basic examples of common equation types:
- Linear Equation in One Variable: 3x - 9 = 0
- Linear Equation in Two Variables: 2x + 5y = 10
- Quadratic Equation: x² - 5x + 6 = 0
- Cubic Equation: x³ - 2x² + x - 2 = 0
- Trigonometric Equation: 2 cos(θ) = 1
4. What is the key difference between a mathematical equation and an expression?
The key difference lies in the presence of an equals sign (=). An equation is a complete statement that shows two expressions are equal (e.g., 5x + 2 = 12). Its purpose is to state a condition of equality, which can then be solved. In contrast, an expression is just a combination of numbers, variables, and operators that represents a single value (e.g., 5x + 2). It does not assert equality and cannot be 'solved' on its own, only simplified or evaluated.
5. How are mathematical equations used to solve real-world problems?
Mathematical equations are fundamental tools for modelling real-world situations. They translate a problem from descriptive language into a mathematical format that can be solved. For example:
- In finance, equations are used to calculate compound interest, loan payments, and investment growth.
- In physics, equations of motion (like F = ma) describe the relationship between force, mass, and acceleration.
- In business, equations help in determining the break-even point, calculating profit and loss, and optimising supply chains.
6. Why is it important to identify the type of an equation before trying to solve it?
Identifying the type of an equation is a critical first step because each type requires a specific set of rules and methods for its solution. For example:
- The method for isolating a variable in a linear equation is completely different from using the quadratic formula for a quadratic equation.
- Solving a trigonometric equation requires knowledge of trigonometric identities and periodicity, which are irrelevant for polynomial equations.
7. What does it mean to 'verify' a solution to an equation, and why is this step crucial?
To 'verify' a solution means to substitute the value you found for the variable back into the original equation. The step is crucial because it confirms the accuracy of your solution. After substitution, you must check if the Left-Hand Side (LHS) of the equation equals the Right-Hand Side (RHS). If they are equal, your solution is correct. This final check helps catch any calculation errors made during the solving process and ensures the integrity of the answer.
8. What is the role of variables and constants in an equation?
In an equation, variables and constants play distinct roles.
- A constant is a known, fixed value that does not change within the equation (e.g., the '5' or '10' in x + 5 = 10).
- A variable (like 'x' or 'y') is a symbol for an unknown quantity whose value we want to find. It can represent a range of values, but within a specific equation, solving it means finding the particular value of the variable that makes the statement of equality true.
