CBSE Class 10 Maths: Pair of Linear Equations — Notes 2026

Introduction and Conditions

• Two equations a₁x+b₁y+c₁=0 and a₂x+b₂y+c₂=0; solution is pair (x,y) satisfying both
• Consistent (unique solution): a₁/a₂ ≠ b₁/b₂; lines intersect at one point
• Inconsistent (no solution): a₁/a₂ = b₁/b₂ ≠ c₁/c₂; parallel lines; Dependent (infinite solutions): a₁/a₂ = b₁/b₂ = c₁/c₂

Graphical Method

• Plot both lines on graph; point of intersection = solution
• To draw line: find two points (x-intercept: y=0; y-intercept: x=0)
• CBSE: describe nature of lines based on condition (intersecting, parallel, coincident)

Substitution Method

• From one equation, express one variable in terms of other; substitute in second equation
• Example: x + y = 10 and x − y = 2; from eq 1: x = 10−y; substitute: (10−y)−y = 2 → y = 4, x = 6
• Best when one equation has simple coefficient (1 or −1)

Elimination Method

• Multiply equations to make coefficients of one variable equal; then add or subtract to eliminate that variable
• Example: 2x+3y=11 and x+2y=7; multiply 2nd by 2: 2x+4y=14; subtract: y=3, substitute to get x=1
• More efficient for larger coefficients; CBSE prefers this method in 3-mark questions

Cross Multiplication Method

• x = (b₁c₂−b₂c₁)/(a₁b₂−a₂b₁); y = (c₁a₂−c₂a₁)/(a₁b₂−a₂b₁)
• Set up cross-multiplication grid and read off numerators and denominator
• Useful when quick answer needed; must memorize formula correctly

Word Problems

• Age problems: present ages from two conditions → two equations; solve
• Number problems: digits problem → ten's digit × 10 + unit's digit = number; condition gives second equation
• Mixture problems: x litres of solution A + y litres of B → amount of component equation; total volume equation

CBSE Exam Tips

• Pair of Linear Equations: 6–8 marks in CBSE; both substitution/elimination and word problems tested
• Word problems: form equations correctly first; solve; verify solution in both original equations
• Graphical method question: use graph paper if provided; label intersection point clearly