Linear Equations in Two Variables

A linear equation in two variables is an equation of the form ax + by + c = 0, where a, b, and c are real numbers, and x and y are variables. The graph of such an equation is always a straight line.

Key Concepts

• Standard form: ax + by + c = 0
• Solution: Any pair (x, y) that satisfies the equation.
• Infinitely many solutions: For every value of x, there is a corresponding value of y.
• Graph: The set of all solutions forms a straight line on the coordinate plane.

Examples

• 2x + 3y = 6
• x - y + 4 = 0
• y = 5x - 7

How to Find Solutions

1. Choose any value for x.
2. Substitute it into the equation to find the corresponding value of y.
3. Each pair (x, y) you find is a solution.

Graphing a Linear Equation

1. Find at least two solutions (points) for the equation.
2. Plot these points on the coordinate plane.
3. Draw a straight line passing through them. This line represents all solutions.

Practice Problems

1. Find three solutions for the equation 3x + 2y = 12.
2. Plot the graph of x + y = 5.
3. Does the point (2, 1) satisfy the equation 4x - y = 7?
4. Write the equation of a line passing through (0, 3) and (2, 7).

Word Problem Example

Summary

• Linear equations in two variables have infinitely many solutions.
• Their graph is always a straight line.
• Each solution is a point on the line.
• They are useful in solving real-life problems involving two quantities.