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LINEAR EQUATIONS IN TWO VARIABLES –
CLASS 10 NOTES
INTRODUCTION
In mathematics, an equation that can be written in the form ax + by + c = 0, where a, b, and c are real numbers and both a and b are not zero, is called a linear equation in two variables. These types of equations represent straight lines when plotted on a Cartesian plane. They are called “linear” because the highest degree of the variables x and y is 1. Such equations have infinitely many solutions, and each solution is an ordered pair (x, y).
GENERAL FORM AND EXAMPLES
The general form of a linear equation in two variables is: ax + by + c = 0 Where: x and y are the variables a, b, and c are real numbers a and b are not both zero
Examples:
3x + 2y = 6 x - y + 4 = 0
2x + 5y - 7 = 0 Each of these represents a straight line on a graph.
SOLUTION OF A LINEAR EQUATION
A solution of a linear equation in two variables is a pair of values (x, y) that satisfies the equation. For example, in the equation 2x + 3y = 12: If x = 0 → y = 4 → (0, 4) is a solution If y = 0 → x = 6 → (6, 0) is a solution There are infinitely many such solutions.
GRAPHICAL REPRESENTATION
To graph a linear equation:
- Choose at least two values for x , find corresponding y values.
- Plot the ordered pairs (x, y) on the Cartesian plane.
- Join the points using a straight line.
Example:
For the equation x + y = 5: x = 0 → y = 5 → point (0, 5) y = 0 → x = 5 → point (5, 0) Join these points to draw the graph.
PAIR OF LINEAR EQUATIONS IN TWO VARIABLES
CONDITIONS FOR CONSISTENCY
Based on the coefficients:
1. Consistent System (has solutions)
Unique Solution: a1/a2 =J b1/b Infinite Solutions: a1/a2 = b1/b2 = c1/c
2. Inconsistent System (no solution)
a1/a2 = b1/b2 =J c1/c
REAL-LIFE APPLICATIONS
Linear equations in two variables are used in: Business (profit & loss, cost) Age problems Mixtures (concentration of solutions) Speed and distance Geometry (lines, angles)
PRACTICE PROBLEMS
Solve: 3x + 4y = 10 and 2x - y = 1 Draw the graph of: x - 2y = 4 Determine the nature of solutions for: 2x + 3y = 12 and 4x + 6y = 24 The sum of two numbers is 15 and their difference is 3. Find the numbers using linear equations.
SUMMARY
A linear equation in two variables has the form: ax + by + c = 0 It represents a straight line. It has infinitely many solutions. A pair of linear equations can have: o One solution (intersecting) o No solution (parallel) o Infinite solutions (same line) Methods of solving: Graphical, Substitution, Elimination, Cross-Multiplication.
