4.1 Introduction
- An equation is a mathematical statement that two expressions are equal.
- Linear equations in two variables involve two variables (x and y) and can be written in the form:
ax+by+c=0
where a ≠ 0, b ≠ 0 and c is constant.
Goal: Find pairs (x, y) that satisfy the equation.
4.2 Linear Equations
- Examples:
- x+y=5
- 2x−3y+6=0
Key Features:
- Graph of linear equation → straight line
- Every point on the line → solution of the equation
Forms of linear equations:
- General form: ax + by + c = 0
- Slope-intercept form: y = mx + c
- m = slope of line
- c = y-intercept
4.3 Solution of a Linear Equation
- A solution of a linear equation in two variables is an ordered pair (x, y) that satisfies the equation.
Methods to find solution:
- Substitution method
- Substitute a value of x or y and find the other variable.
- Example: x + y = 5 → if x = 2, y = 3 → (2, 3) is solution.
- Graphical method
- Draw the line on the coordinate plane.
- Any point on the line is a solution.
- Elimination method (used when solving system of two equations)
- Combine two equations to eliminate one variable.
Example:
Equation: 2x + y = 6
- Let x = 0 → y = 6 → (0, 6)
- Let x = 3 → y = 0 → (3, 0)
- Plot points (0,6) and (3,0) → Draw line → All points on line are solutions
Quick Short Q&A (Most Possible)
| Question | Short Answer |
|---|---|
| Form of linear equation in two variables? | ax + by + c = 0 |
| Solution of linear equation? | Ordered pair (x, y) satisfying equation |
| Graph of linear equation? | Straight line |
| x + y = 5, x=2 → y=? | 3 |
| Methods to solve linear equations? | Substitution, Graphical, Elimination |
| Slope-intercept form? | y = mx + c |
| What is slope? | Inclination of line |
| y-intercept? | Point where line meets y-axis |
| If x = 0 in 2x + y = 6 → y=? | 6 |
| Any point on line? | Solution of equation |