source: ncert
topic: algebra
class_level: class_9
chapter: ch4
difficulty: beginner
Class 9 | Ch4: Linear Equations in Two Variables FORM: ax+by+c=0 where a,b are not both zero. x and y are variables. A linear equation in two variables has INFINITELY MANY solutions. Each solution is an ordered pair (x,y) that satisfies the equation. GRAPH OF LINEAR EQUATION IN TWO VARIABLES: Graph is always a STRAIGHT LINE. To draw: find at least 2 solutions (3 recommended to verify). For 2x+3y=12: x=0: 3y=12, y=4. Point (0,4). y=0: 2x=12, x=6. Point (6,0). x=3: 6+3y=12, y=2. Point (3,2). All 3 collinear confirms straight line. X-INTERCEPT: point where line crosses x-axis. Set y=0 and solve for x. Y-INTERCEPT: point where line crosses y-axis. Set x=0 and solve for y. EQUATIONS OF SPECIAL LINES: x=a: vertical line parallel to y-axis, passing through (a,0). y=b: horizontal line parallel to x-axis, passing through (0,b). y=x: line through origin, bisects I and III quadrants. EXPRESSING WORD PROBLEMS AS LINEAR EQUATIONS: "Cost of x pencils at 2 each and y pens at 5 each is 50": 2x+5y=50. SOLVED EXAMPLES: Example 1: Draw graph of 2x+3y=12. Find x and y intercepts. y-intercept (x=0): y=4. So (0,4). x-intercept (y=0): x=6. So (6,0). Plot (0,4), (6,0), (3,2). Join them. Example 2: Find 3 solutions of x+2y=8. (0,4), (8,0), (2,3). Verify: 0+8=8, 8+0=8, 2+6=8. All correct. COMMON MISTAKES: A linear equation in two variables has infinite solutions (not just 2 or 3). Every point on the graph is a solution. Every solution gives a point on the graph. Graph of x=3 is a VERTICAL line (not a point on x-axis).