Chapter 13 — Two Variables, One Line

Chapter 13 of the Class 9 Maths NCERT textbook (Ganita Manjari), "Two Variables, One Line", introduces linear equations in two variables, their solutions as ordered pairs, slope and the slope-intercept form, and methods for solving a pair of such equations algebraically and graphically. Students gain the tools to set up, solve, and interpret systems of linear equations drawn from real-life contexts like pricing, ages, and digit puzzles.

• Standard Form and Solutions — A linear equation in two variables takes the standard form ax + by + c = 0, where a and b are not both zero. Any value assigned to x yields a corresponding y, producing infinitely many solutions, each written as an ordered pair (x, y).
• Geometric Meaning of a Line — Every solution of a linear equation corresponds to a point on its graph, and all such points lie on a single straight line. Two intercept points — found by setting x = 0 and y = 0 — are enough to draw this line.
• Slope and Slope-Intercept Form — The slope of a line through (x1, y1) and (x2, y2) is (y2 − y1)/(x2 − x1), measuring steepness as rise over run. In the form y = mx + d, m is the slope and d is the y-intercept where the line crosses the y-axis.
• Solving a Pair of Equations — Two linear equations can be solved algebraically by substitution (express one variable, substitute into the other equation) or by elimination (multiply equations to equalise one coefficient, then add or subtract). A pair may have a unique solution, infinitely many solutions, or no solution.
• Graphical Solution and Nature of Lines — Graphing both equations reveals the nature of the system: intersecting lines give a unique solution, coincident lines give infinitely many solutions, and parallel lines give no solution — determined by the ratios a1/a2, b1/b2, and c1/c2.

1. 01Standard form of a linear equation in two variables: ax + by + c = 0, where a and b are not both zero.
2. 02A linear equation in two variables has infinitely many solutions, each expressed as an ordered pair (x, y).
3. 03Slope = Rise / Run = (y2 − y1) / (x2 − x1); it is the same for any two points on the line.
4. 04In y = mx + d, m is the slope and d is the y-intercept (the point (0, d) on the y-axis).
5. 05When c = 0 in ax + by + c = 0, the line passes through the origin.
6. 06Unique solution when a1/a2 ≠ b1/b2; infinitely many when a1/a2 = b1/b2 = c1/c2; no solution when a1/a2 = b1/b2 ≠ c1/c2.
7. 07Two algebraic methods: Substitution (express one variable, substitute) and Elimination (equalise one coefficient, add or subtract).