Chapter at a glance
Chapter 13 builds on Linear Polynomials (Part I, Ch 2). You learn the standard form of a linear equation in two variables, what its solutions look like (a straight line), the slope and the slope-intercept form, and three ways to solve a pair of linear equations: substitution, elimination and graphs.
13.1 Linear equations in two variables
Radha buys x kg of mangoes at ₹60/kg and y kg of bananas at ₹50/kg and pays ₹280: 60x + 50y = 280.
Standard form
ax + by + c = 0, where a, b, c are real numbers and a and b are not both zero. a and b are the coefficients, c is the constant.
- Fractions are fine, but we usually clear them: (1/4)x + (5/3)y − 2 = 0 becomes 3x + 20y − 24 = 0 after multiplying by 12.
- ax + c = 0 is a special case with b = 0: 7 + 4x = 0 is 4x + 0·y + 7 = 0.
- The same equation has many standard forms: 5x + 2y − 2.7 = 0 or −5x − 2y + 2.7 = 0.
13.2 Solutions and their graph
A solution is an ordered pair (x, y) that satisfies the equation. For 3x + 2y = 12: (2, 3) and (4, 0) are solutions; (1, 3) is not, since 3 + 6 = 9 ≠ 12.
Infinitely many solutions
Give x any value u and solve the resulting one-variable equation for y. So every linear equation in two variables has infinitely many solutions, and when plotted they form a straight line. Every point on the line is a solution, and every solution is a point on the line.
Quickest way to draw the line
Put x = 0 to find where it cuts the y-axis and y = 0 to find where it cuts the x-axis. For 4x + 3y = 12: (0, 4) and (3, 0). If c = 0 the line passes through the origin; then pick another x (for 2x + 5y = 0: (0, 0) and (5, −2)).
Finding unknown coefficients: if (−3, 4) satisfies 3ax + 4y = −2 and 2x + by = 14, substitute to get −9a + 16 = −2, so a = 2, and −6 + 4b = 14, so b = 5.
Multiplying an equation by k ≠ 0 does not change its solutions: 3x + 4y = 7 and 6x + 8y = 14 have the same solution set.
13.3 Slope and slope-intercept form
Slope (gradient)
slope = Rise ÷ Run = (y₂ − y₁) ÷ (x₂ − x₁)
It is the same for any two points on a line, and the order of the points does not matter. Through (−4, 5) and (−1, 2): (2 − 5) ÷ (−1 + 4) = −1.
Slope-intercept form: y = mx + d
m = slope, d = y-intercept (the line cuts the y-axis at (0, d)).
Why m is the slope: if (x₁, y₁) and (x₂, y₂) lie on y = mx + d, then (y₂ − y₁) ÷ (x₂ − x₁) = m(x₂ − x₁) ÷ (x₂ − x₁) = m.
From standard form (b ≠ 0): y = −(a/b)x − c/b, so m = −a/b and d = −c/b.
| Slope | Line | Example |
|---|---|---|
| m > 0 | Rises left to right | y = 2x + 1: up 2 for every 1 to the right |
| m < 0 | Falls left to right | y = −3x + 4: down 3 for every 1 to the right |
| m = 0 | Horizontal, parallel to the x-axis | y = 5 |
| Undefined | Vertical, parallel to the y-axis (run = 0) | x = 2 (cannot be written as y = mx + d) |
Ramp example: a wheelchair ramp needs 1 cm rise per 12 cm run. For 18 cm of stairs the run must be 18 × 12 = 216 cm.
13.4–13.5 Solving a pair of linear equations
A pair: a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0. A solution is an ordered pair satisfying both. The idea is to reduce the pair to one equation in one variable.
Substitution method
- From one equation, express one variable in terms of the other.
- Substitute into the other equation and solve.
- Substitute back to get the second variable, then verify in both equations.
Example: 7x − 15y = 2 and x + 2y = 3. Then x = 3 − 2y, so 7(3 − 2y) − 15y = 2 and −29y = −19, giving y = 19/29, x = 49/29.
Example: triangle angles x°, y° = 4x° and 50°: x + 4x + 50 = 180, so x = 26° and y = 104°.
Elimination method
- Multiply the equations so one variable has equal coefficients.
- Subtract (or add) to eliminate it.
- Solve, then substitute back.
Example (incomes 9 : 7, expenditures 4 : 3, each saves ₹2000): 9x − 4y = 2000 and 7x − 3y = 2000. Multiply by 3 and 4: 27x − 12y = 6000 and 28x − 12y = 8000. Subtracting gives x = 2000, then y = 4000. Incomes are ₹18,000 and ₹14,000.
Example (digits): (10x + y) + (10y + x) = 66 gives x + y = 6; with x − y = 2, x = 4 and y = 2, so the number is 42 (42 + 24 = 66).
The three possible cases
| Ratio test | Graph | Solutions |
|---|---|---|
| a₁/a₂ ≠ b₁/b₂ | Lines intersect at one point | Unique (consistent) |
| a₁/a₂ = b₁/b₂ = c₁/c₂ | Lines coincide | Infinitely many |
| a₁/a₂ = b₁/b₂ ≠ c₁/c₂ | Lines are parallel | None (inconsistent) |
Examples: x − y = 10 and 10x − 10y = 100 give infinitely many; x − y = 10 and 10x − 10y = 101 give none. With elimination, if both variables vanish you get 0 = 0 (infinite) or something like 0 = 1 (none).
13.6 Graphical method
Draw both lines using two points each. The intersection point is the solution.
- x + 3y = 6 through (0, 2), (6, 0) and 2x − 3y = 12 through (0, −4), (3, −2) meet at (6, 0): unique solution.
- x + 2y − 4 = 0 and 2x + 4y − 12 = 0: ratios 1/2 = 2/4 ≠ 4/12, so the lines are parallel with no solution. Lines with equal slope and different intercepts are parallel.
- 2x + 3y = 9 and 4x + 6y = 18 (Romila and Sonali’s erasers and sheets): every ratio is 1/2, so the lines coincide.
A pinch of history: Babylonian tablets (c. 2000 BCE) have linear systems. China’s Nine Chapters on the Mathematical Art devotes the fangcheng chapter to elimination using counting rods, eighteen centuries before Gauss. Āryabhaṭa (499 CE) and Brahmagupta (628 CE) also solved systems by elimination.
Practice questions with answers
- Mahāvīrāchārya (c. 850 CE): 9 citrons + 7 wood-apples cost 107; 7 citrons + 9 wood-apples cost 101. Find each price. Add: 16(x + y) = 208, so x + y = 13. Subtract: 2(x − y) = 6, so x − y = 3. Citron 8, wood-apple 5.
- 5 pencils + 7 pens = ₹50; 7 pencils + 5 pens = ₹46. x + y = 8 and y − x = 2, so pencil ₹3, pen ₹5.
- Taxi: ₹155 for 10 km and ₹220 for 15 km. Fixed charge, per-km rate, and fare for 25 km? Per km ₹13, fixed ₹25, 25 km costs ₹350.
- Plan A: ₹50 + ₹0.20/min. Plan B: ₹30 + ₹0.30/min. When are they equal? 50 + 0.2m = 30 + 0.3m gives m = 200 min. B is cheaper below 200 min, A above.
- Use ratios: 9x + 3y + 12 = 0 and 18x + 6y + 24 = 0. All ratios are 1/2: coincident, infinitely many solutions.
- 6x − 3y + 10 = 0 and 2x − y + 9 = 0. 6/2 = −3/−1 = 3 but 10/9 ≠ 3: parallel, no solution.
- Half the perimeter of a garden is 36 m and its length is 4 m more than its width. l + w = 36, l − w = 4, so 20 m × 16 m.
- Give a formula for the x-intercept of y = mx + c (m ≠ 0). Put y = 0: x = −c/m.
Chapter summary
- ax + by + c = 0 (a, b not both 0) is a linear equation in two variables, with infinitely many solutions.
- y = mx + d: m is the slope and (0, d) is the y-intercept. Slope = (y₂ − y₁)/(x₂ − x₁).
- Points on the graph are exactly the solutions.
- Solve pairs by substitution, elimination or graphing. Use the ratios a₁/a₂, b₁/b₂, c₁/c₂ to predict a unique solution, no solution or infinitely many.
Source: NCERT, Ganita Manjari, Grade 9 (2026-27). NCERT chapter PDF. Spotted a mistake? Email edura.class9.yt@gmail.com. Last updated 11 October 2026.