Class 10 Maths Notes — Chapter 3: Pair of Linear Equations in Two Variables
Systems of two linear equations — geometric interpretation, algebraic solution methods, and word problems.
Detailed NCERT notes
- General form: a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0.
- Graphical meaning of a linear equation in two variables = a straight line. A pair of equations = two lines. Solutions = points of intersection.
- Three cases based on ratios:
- (i) a₁/a₂ ≠ b₁/b₂ → lines intersect at exactly one point → consistent, unique solution.
- (ii) a₁/a₂ = b₁/b₂ = c₁/c₂ → lines coincide → consistent, infinitely many solutions (dependent).
- (iii) a₁/a₂ = b₁/b₂ ≠ c₁/c₂ → lines parallel → inconsistent, no solution.
- Algebraic methods: (1) Substitution, (2) Elimination, (3) Cross-multiplication.
- Substitution method: express one variable from one equation, plug into the other.
- Elimination method: multiply equations to align coefficients, add/subtract to eliminate one variable.
- Cross-multiplication method: x/(b₁c₂ − b₂c₁) = y/(c₁a₂ − c₂a₁) = 1/(a₁b₂ − a₂b₁).
- Some non-linear equations reduce to linear form by substitution: e.g. 1/x = u, 1/y = v.
- Word problems: identify two unknowns, form two equations from two conditions, solve algebraically.
- Typical word-problem types: age problems, digits of a two-digit number, fraction problems, speed/time/distance (boat & stream, upstream/downstream), work-time, geometry-based (angles of triangle, sides).
- For two-digit number problems, if digits are x (tens) and y (units), number = 10x + y and reverse = 10y + x.
- Boat problems: speed downstream = boat + current, upstream = boat − current.
Formulas & key results
- Consistent unique: a₁/a₂ ≠ b₁/b₂
- Consistent dependent (∞): a₁/a₂ = b₁/b₂ = c₁/c₂
- Inconsistent: a₁/a₂ = b₁/b₂ ≠ c₁/c₂
- Cross-multiplication: x/(b₁c₂ − b₂c₁) = y/(c₁a₂ − c₂a₁) = 1/(a₁b₂ − a₂b₁)
Mind map
- Graphical → substitution → elimination → cross-multiplication
- Ratio test for nature of solutions
- Reducible to linear form (1/x, 1/y)
- Word problems: age, digits, fraction, boat, work
Tricks & shortcuts
- Elimination when coefficients match easily.
- Substitution when a variable is already isolated.
- Check ratios first when asked about nature of solutions.
Common mistakes to avoid
- Sign errors in cross-multiplication.
- Forgetting to check c₁/c₂ for dependency.
- Setting up reverse two-digit number as 10x + y instead of 10y + x.
Competency-based questions & answers
- Q. Father is 3 times his son. In 12 years, father is 2× son. Find their ages.A. F = 3S; F + 12 = 2(S + 12) ⇒ 3S + 12 = 2S + 24 ⇒ S = 12, F = 36.
- Q. For what value of k does 2x + 3y = 7 and (k − 1)x + (k + 2)y = 3k have infinite solutions?A. 2/(k−1) = 3/(k+2) = 7/(3k) ⇒ solve: k = 7.
- Q. A boat covers 30 km downstream in 3 h and 30 km upstream in 5 h. Find boat and stream speeds.A. b + s = 10, b − s = 6 ⇒ b = 8 km/h, s = 2 km/h.