Pair of Linear Equations in Two Variables

A linear equation in two variables is an equation of the form:

1. Graphical Method of Solution

Steps:

2. Algebraic Methods of Solution

There are three main methods:

A. Substitution Method

1. Solve one equation for one variable.

2. Substitute in the second equation.

3. Solve for the remaining variable.

4. Back-substitute to find the first variable.

B. Elimination Method

1. Multiply one or both equations (if necessary) so coefficients of a variable are equal.

2. Add or subtract equations to eliminate a variable

3. Solve for remaining variable.

4. Back-substitute to find the other variable.

C. Cross-Multiplication Method

3. Types of Solutions

1. Unique Solution: Lines intersect at one point → Consistent and independent

2. No Solution: Lines are parallel → Inconsistent.

3. Infinite Solutions: Lines coincide → Consistent and dependent

4. Condition for Solvability

5. Word Problems

1. Age problems

2. Number of items or money

3. Distance-speed-time

4. Mixture problems

Example 5:

A fruit seller sold 50 apples and 30 oranges for ₹350. Another day he sold 30 apples and 20 oranges for ₹210. Find the cost of each apple and orange.

• Let apple = ₹x, orange = ₹y

• Equations:

6. Summary Table

7. Exercises for Practice

This chapter explanation is complete, sequential, and includes definitions, methods, examples, word problems, and exercises just like a textbook.