Chapter 3 — Pair of Linear Equations in Two Variables

NCERT Class 10 Maths Chapter 3 covers Pair of Linear Equations in Two Variables, teaching students to solve such pairs using graphical, substitution, and elimination methods, and to determine whether a pair is consistent, inconsistent, or dependent based on the ratio of coefficients.

Chapter 3 of NCERT Class 10 Mathematics introduces pairs of linear equations in two variables and their solutions. Two lines on a graph can intersect at a single point (unique solution, consistent pair), coincide (infinitely many solutions, dependent and consistent), or be parallel (no solution, inconsistent pair). The chapter covers three solution methods: graphical method, substitution method, and elimination method. The nature of solutions is determined by comparing the ratios a1/a2, b1/b2, and c1/c2 of the coefficients of the two equations.

1. 01A pair of linear equations can be consistent (has at least one solution) or inconsistent (has no solution); a dependent pair is always consistent with infinitely many solutions.
2. 02Graphically, two lines either intersect at one point (unique solution), coincide (infinite solutions), or are parallel (no solution).
3. 03The substitution method involves expressing one variable in terms of the other and substituting into the second equation to solve.
4. 04The elimination method multiplies equations by suitable constants to make coefficients of one variable equal, then adds or subtracts to eliminate that variable.
5. 05If a1/a2 ≠ b1/b2, the pair is consistent with a unique solution; if a1/a2 = b1/b2 = c1/c2, it is dependent (infinitely many solutions); if a1/a2 = b1/b2 ≠ c1/c2, it is inconsistent (no solution).
6. 06Real-life problems such as age puzzles, cost problems, and digit problems can be modelled and solved as pairs of linear equations.