Mathematics Notes for Class 11 Linear Equations-In-Two-Variables

Linear Equations in Two Variables

Welcome to the chapter on Linear Equations in Two Variables for Class 11. In this chapter, you will learn what linear equations are, how to represent them graphically, and how to solve them using different methods. By the end of this chapter, you will be able to solve real-life problems using linear equations in two variables.

Key Concepts

Linear Equation: An equation of the form ax + by + c = 0, where a, b, and c are real numbers and a and b are not both zero.
Solution: A pair of values (x, y) that satisfies the equation.
Graph: The set of all points (x, y) that satisfy the equation forms a straight line on the coordinate plane.

General Form

The general form of a linear equation in two variables is:

ax + by + c = 0

a and b are coefficients of x and y.
c is a constant.

Graphical Representation

The graph of a linear equation in two variables is a straight line. Every point on the line is a solution to the equation.

To draw the graph, find at least two solutions (points) and plot them on the coordinate plane. Join the points to get the line.
Example: For the equation x + y = 4, points (0, 4) and (4, 0) satisfy the equation.

Solving Linear Equations in Two Variables

To solve a pair of linear equations in two variables, you can use:

Graphical Method: Plot both equations on the graph. The point where the lines intersect is the solution.
Substitution Method: Solve one equation for one variable and substitute in the other.
Elimination Method: Add or subtract the equations to eliminate one variable.

Example

Solve the following system:

2x + 3y = 12
x - y = 1

Solution (by substitution):
From the second equation: x = y + 1
Substitute in the first: 2(y + 1) + 3y = 12 ⇒ 2y + 2 + 3y = 12 ⇒ 5y = 10 ⇒ y = 2
So, x = 2 + 1 = 3
Solution: (x, y) = (3, 2)

Applications

Solving real-life problems involving two unknowns, such as age, money, distance, etc.
Used in business, science, and engineering to model relationships between quantities.

Practice Questions

Find two solutions of the equation 3x + 2y = 12.
Draw the graph of x - 2y = 4.
Solve the system: x + y = 7 and x - y = 3.
If 2x + y = 10 and x - y = 2, find the values of x and y.
Explain the graphical method of solving two linear equations in two variables.

Challenge Yourself

Create your own pair of linear equations and solve them using the elimination method.
Find the point of intersection of the lines 2x - y = 1 and x + y = 5 graphically.

Did You Know?

Linear equations are used in computer graphics to draw straight lines on screens.
Systems of linear equations can have one solution, no solution, or infinitely many solutions.

Glossary

Linear Equation: An equation whose graph is a straight line.
Variable: A symbol (like x or y) that stands for a number.
Solution: The value(s) of the variable(s) that make the equation true.
Graph: A picture that shows all the solutions of an equation.

Answers to Practice Questions

x = 2, y = 3 (3×2 + 2×3 = 6 + 6 = 12); x = 4, y = 0 (3×4 + 2×0 = 12 + 0 = 12)
Plot points like (6,1) and (8,2) and join them to draw the line x - 2y = 4.
Add: (x + y) + (x - y) = 7 + 3 ⇒ 2x = 10 ⇒ x = 5; y = 2
x = 4, y = 2
Draw both lines on the graph. The point where they meet is the solution.

Practice solving linear equations to build a strong foundation in algebra and real-world problem solving!

Mathematics Class 11 - Linear Equations-In-Two-Variables Notes