Learning Objectives
- Understand linear equations in two variables
- Find solutions of a linear equation in two variables
- Graph a linear equation on the Cartesian plane
- Understand equations of lines parallel to the x-axis and y-axis
Key Concepts
Linear Equation in Two Variables
An equation of the form ax + by + c = 0, where a, b, c are real numbers and both a and b are not zero, is called a linear equation in two variables x and y.
Examples: 2x + 3y = 12, x - y + 5 = 0
Solution of a Linear Equation
A solution is an ordered pair (x, y) that satisfies the equation. A linear equation in two variables has infinitely many solutions.
Example: For 2x + y = 6, solutions include (0, 6), (1, 4), (2, 2), (3, 0), etc.
Graph of a Linear Equation
The graph of every linear equation in two variables is a straight line. Every point on the line is a solution of the equation, and every solution of the equation is a point on the line.
To draw the graph: find at least two solutions, plot them, and join with a straight line.
Equations of Lines Parallel to Axes
y = k (constant): A line parallel to the x-axis at a distance k from it.
x = k (constant): A line parallel to the y-axis at a distance k from it.
y = 0: The equation of the x-axis.
x = 0: The equation of the y-axis.
Important Forms
Any linear equation in one variable, say x = a, can be written as a linear equation in two variables: 1·x + 0·y = a. Its graph is a vertical line.
Similarly, y = b can be written as 0·x + 1·y = b. Its graph is a horizontal line.
Summary
A linear equation in two variables is of the form ax + by + c = 0 and has infinitely many solutions. Its graph is always a straight line. Lines parallel to the x-axis have the form y = constant, and lines parallel to the y-axis have the form x = constant.
Important Terms
- Linear Equation in Two Variables: Equation of the form ax + by + c = 0 where a ≠ 0 or b ≠ 0
- Solution: An ordered pair (x, y) that makes the equation true
- Graph: The set of all points (x, y) that satisfy the equation, forming a straight line
Quick Revision
- ax + by + c = 0 is the general form of a linear equation in two variables
- A linear equation in two variables has infinitely many solutions
- The graph is always a straight line
- y = k is a horizontal line; x = k is a vertical line
- x = 0 is the y-axis; y = 0 is the x-axis