Linear Equations in Two Variables — Mathematics Class 9 Notes (CBSE & HBSE)
Free NCERT Mathematics notes for Linear Equations in Two Variables (Class 9) on Siksha Sarovar, aligned to CBSE and Haryana Board (HBSE). This chapter is broken into 3 topics with clear explanations, formulas, solved examples and board-pattern practice — free to read, no sign-up required.
Board exam focus — Linear Equations in Two Variables (CBSE & HBSE)
Bridges algebra and geometry — every solution is now a point on the line, not just a number. CBSE and HBSE both ask graph-based and word-problem questions worth 5–6 marks.
Standard Form & Solutions
Standard Form
ax + by + c = 0, where a, b, c are real numbers and a, b are not both zero.
Every linear equation has infinitely many solutions, because for any chosen x there is a corresponding y (and vice-versa).
How to Find Solutions
Pick convenient x-values (often 0, 1, 2, −1) and solve for y.
Example: 2x + y = 7
| x | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| y | 7 | 5 | 3 | 1 |
Each (x, y) pair satisfies the equation.
Equations of Lines Parallel to Axes
- y = k ⇒ horizontal line (parallel to x-axis).
- x = h ⇒ vertical line (parallel to y-axis).
- y = 0 is the x-axis itself; x = 0 is the y-axis.
Verifying a Solution
Substitute the pair into the equation; if LHS = RHS, the point is a solution.
CBSE Tip: "Infinitely many" must explicitly appear in your answer when asked about number of solutions — single-line responses scoring 1 mark hinge on this exact phrasing.
Graph of a Linear Equation
Plotting the Graph
- Write three solution pairs (use a table).
- Plot them on the Cartesian plane.
- Join — the three points should be collinear. If not, recheck arithmetic.
- Extend the line both directions and add arrowheads.
Why a Line?
A linear equation in two variables represents a line in the plane. Every point on that line is a solution; every solution lies on the line.
Reading Intercepts
- Put x = 0 ⇒ get y-intercept.
- Put y = 0 ⇒ get x-intercept.
Example: 2x + 3y = 12.
- x = 0 ⇒ y = 4 ⇒ (0, 4).
- y = 0 ⇒ x = 6 ⇒ (6, 0).
Recognising Special Lines on Sight
| Equation | Graph |
|---|---|
| y = 5 | horizontal at height 5 |
| x = −2 | vertical at x = −2 |
| y = x | line through origin, 45° |
| y = −x | line through origin, 135° |
HBSE Tip: Always use three points before joining — a two-point graph cannot catch a calculation error.
Real-Life Word Problems
Modelling Steps
- Define the two unknowns (let x = …, y = …).
- Translate the condition into a linear equation.
- Write in standard form ax + by + c = 0.
- (If asked) plot two or three solutions; verify graphically.
Common Templates
| Real condition | Equation |
|---|---|
| "Cost of 2 pens + 3 books = ₹70" | 2x + 3y = 70 |
| "Twice age of son + father's age = 80" | 2x + y = 80 |
| "Sum of digits of two-digit number = 9" | x + y = 9 |
Distance, Speed, Time (Linear Version)
If a car covers d km at a constant speed v in t hours, then d = vt — for fixed v this is a linear equation in d and t.
Currency Conversion
If 1 USD = ₹83 (constant rate), then y = 83x is a linear equation through origin.
CBSE Tip: After writing the equation, mark in your answer the meaning of x and y. Marks are awarded for each clear definition.
Frequently asked questions
Are these Linear Equations in Two Variables notes free?
Yes — the Linear Equations in Two Variables notes for Mathematics (Class 9) on Siksha Sarovar are completely free to read, with no account required.
Do these notes follow CBSE and HBSE?
Yes. The Linear Equations in Two Variables notes are NCERT-aligned and include guidance for both CBSE and Haryana Board (HBSE), with important questions and MCQs for revision.
What does the Linear Equations in Two Variables chapter cover?
Concept explanations, key formulas and definitions, fully solved examples and board-pattern practice questions for Linear Equations in Two Variables.