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Linear Equations in Two Variables is an important chapter in Class 9 maths syllabus that helps students understand the relationship between two quantities. This chapter explains how to form linear equations, represent them on a graph, and find their solutions in a simple and logical way. Learning this chapter builds a strong base for algebra and helps students apply mathematical concepts to real-life situations.
Linear Equations In Two Variables Class 9 Notes PDF
These notes on Class 9 Linear Equations in Two Variables are designed to make the chapter easy and understandable for students. All key concepts, formulas, and examples are explained in a clear and simple manner. Download the complete notes from below to revise effectively and score better.
1. Linear Equation in Two Variable Introduction
The chapter Linear Equations in One Variable teaches you to solve easy algebraic equations through simple mathematical operations. The equations aid you to determine unknown values (such as 'x') in age, money, number, and geometry problems. You'll study how to push all the terms onto one side, simplify them, and obtain your answer step by step.
This is a good base for math later on, so it is extremely important to understand it well. Whether you're moving terms or doing brackets, once you understand the reasoning, it's really not difficult and fun!
2. Linear Equation
A linear equation is an equation that can be written in the form ax + by + c = 0, where 'a', 'b', and 'c' are real numbers, and 'a' and 'b' are not both zero. These equations represent a straight line when graphed. They can have one or two variables, and the highest power of any variable is one.
Example of linear equation:
=> 2x+ 6=2
- 2x = 2- 6
- 2x = – 4
- x = – 4/2
- x = – 2
This is the example of Linear equation
3. Linear Equation Formula
The linear equation formula is the way of expressing a linear equation. This can be done in different ways.
For example, a linear equation can be expressed in the standard form, the slope-intercept form, or the point-slope form. Now, if we take the standard form of a linear equation, let us learn the way in which it is expressed.
We can see that it varies from case to case based on the number of variables and it should be remembered that the highest (and the only) degree of all variables in the equation should be 1.
Note: The slope of a linear equation is the amount by which the line is rising or falling. It is calculated by the formula rise/run. i.e., if (x1, y1) and (x2, y2) are any two points on a line then its slope is calculated using the formula (y2 - y1)/(x2 - x1).
4. Linear Equations in Standard Form
The standard form or the general form of linear equations in one variable is written as, Ax + B = 0; where A and B are real numbers, and x is the single variable.
The standard form of linear equations in two variables is expressed as, Ax + By = C; where A, B and C are any real numbers, and x and y are the variables.
5. Linear equation in two variables
- An equation of the form, ax + by + c = 0, where a, b and c are constants, such that a and b are both not zero and x and y are variables that are called a linear equation in two variables.
- For example, 2x + 3y + 10 = 0, 3x + 7y = 0. Real life situations can be expressed mathematically as linear equations in two variables.
6. Meaning of Solution
A solution of a linear equation in two variables is a pair of values (x, y) that makes the equation true.
Example:
For the equation x + y = 5
- (1, 4), (2, 3), (0, 5) are solutions
- There are infinitely many solutions.
Infinite Solutions: Unlike linear equations in one variable, a linear equation in two variables has infinitely many solutions because for every value of x, there is a corresponding value of y.
7. Graphical Representation
The graph of a linear equation in two variables is always a straight line.
- Each solution (x, y) is represented as a point on the line.
- All points on the straight line satisfy the equation.
a) How to Draw the Graph?
Steps to draw the graph of a linear equation:
- Write the equation.
- Choose any two values of x and find corresponding values of y.
- Plot the points on graph paper.
- Join the points to get a straight line.
b) Solutions of Linear Equations in Two Variables on a Graph
A linear equation of the form ax + by + c = 0 is represented graphically by a straight line.
Every point lying on this line represents a solution of the linear equation, and conversely, each solution of the linear equation corresponds to a point on the line.
c) Lines Passing Through the Origin
Some linear equations have (0, 0) as a solution. When such equations are drawn on a graph, the resulting straight line passes through the origin.
The coordinate axes themselves are examples of such lines: the x-axis is represented by y = 0, and the y-axis is represented by x = 0.
d) Lines Parallel to the Coordinate Axes
Linear equations of the form y = a are represented by straight lines parallel to the x-axis, where a is the y-coordinate of every point on the line.
Similarly, linear equations of the form x = a are represented by straight lines parallel to the y-axis, where a is the x-coordinate of every point on the line.
Intercepts
- x-intercept: Value of x when y = 0
- y-intercept: Value of y when x = 0
Example:
For 2x + y = 4
- x-intercept = 2
- y-intercept = 4
8. Table of Values Method
A table of values helps to find different solutions of the equation and makes graph plotting easier.
Important Points to Remember
- Linear equations in two variables have no single unique solution.
- Their graph is always a straight line.
- At least two points are required to draw the graph.
- Solutions are written in ordered pairs (x, y).
FAQs
Q1. Why does a linear equation in two variables represent a straight line on a graph?
Ans. Because the highest power of both variables is 1, the relationship between x and y changes at a constant rate. This constant rate of change produces a straight line when plotted on the Cartesian plane.
Q2. Can a linear equation in two variables have more than one solution?
Ans. Yes. It has infinitely many solutions. Every point on the straight line represented by the equation satisfies the equation.
Q3. What is the role of constants a, b, and c in the equation ax + by + c = 0?
Ans. The constants a and b determine the slope and direction of the line, while c determines where the line cuts the axes. Both a and b cannot be zero at the same time.
Q4. How is a linear equation in two variables represented graphically?
Ans. By finding at least two solutions of the equation, plotting the corresponding points on the Cartesian plane, and joining them with a straight line.
Q5. Why do we need at least two points to draw the graph of a linear equation?
Ans. A straight line is uniquely determined by two distinct points. One point alone is not enough to decide the direction of the line.
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