Pair of Linear Equations in Two Variables

Class 10 Maths - Chapter 3 | NCERT Formula Sheet

1. Standard Form

a1x + b1y + c1 = 0
a2x + b2y + c2 = 0

where a1, b1, c1, a2, b2, c2 are real numbers, a1 & b1 not both zero, a2 & b2 not both zero.

2. Types of Solutions

Consistent (One Solution)

Intersecting lines

a1/a2 ? b1/b2

Consistent (Infinite Solutions)

Coincident lines

a1/a2 = b1/b2 = c1/c2

Inconsistent (No Solution)

Parallel lines

a1/a2 = b1/b2 ? c1/c2

3. Algebraic Methods

Substitution Method

  1. Express x (or y) from first equation
  2. Substitute in second equation
  3. Solve for remaining variable
  4. Substitute back to find first variable

Elimination Method

  1. Multiply equations to equalize coefficients
  2. Add/subtract to eliminate one variable
  3. Solve for remaining variable
  4. Substitute back

Cross-Multiplication Method

x/(b1c2 - b2c1) = y/(c1a2 - c2a1) = 1/(a1b2 - a2b1)

4. Solution Formulas

When a1b2 - a2b1 ? 0:
x = (b1c2 - b2c1)/(a1b2 - a2b1)
y = (c1a2 - c2a1)/(a1b2 - a2b1)

5. Graphical Method

  1. Find two points for each equation (use x=0, y=0)
  2. Plot points on graph
  3. Draw straight lines
  4. Check intersection point(s)
• Intersecting: 1 point ? unique solution
• Coincident: infinite points ? infinite solutions
• Parallel: no intersection ? no solution

6. Word Problem Types (NCERT)

  1. Number Problems: Find two numbers
  2. Age Problems: Present/past/future ages
  3. Money Problems: Cost of items
  4. Speed Problems: Distance = Speed × Time
  5. Geometry Problems: Angles, perimeter
  6. Mixture Problems: Ratio & proportion
  7. Work Problems: Time & work together

7. Special Cases

Equations Reducible to Linear Form

Example: 2/x + 3/y = 13
Let p = 1/x, q = 1/y
? 2p + 3q = 13

Symmetric Equations

ax + by = c
bx + ay = d
Solve by adding & subtracting

8. Important Notes

9. Quick Reference Table

Condition

a1/a2 ? b1/b2

Result: Unique solution

Condition

a1/a2 = b1/b2 = c1/c2

Result: Infinite solutions

Condition

a1/a2 = b1/b2 ? c1/c2

Result: No solution

10. Example (NCERT Style)

Problem: Solve 2x + 3y = 8 and 4x + 6y = 7

Solution:
Here a1/a2 = 2/4 = 1/2
b1/b2 = 3/6 = 1/2
c1/c2 = 8/7 ? 1/2
? a1/a2 = b1/b2 ? c1/c2
Hence, equations are inconsistent (no solution)