What are the methods to solve a pair of linear equations?

Four methods: (1) Graphical — plot both lines, intersection is solution; (2) Substitution — express x or y from one equation, substitute in other; (3) Elimination — multiply equations to equalise one coefficient, then add/subtract to eliminate it; (4) Cross-multiplication — direct formula using coefficients. Elimination and substitution are most common in CBSE exams.

How do you check if a pair of linear equations has no solution?

Compare ratios of coefficients: a₁/a₂ = b₁/b₂ ≠ c₁/c₂ → no solution (parallel lines). Example: 2x + 3y = 5 and 4x + 6y = 10: a₁/a₂ = 2/4 = 1/2, b₁/b₂ = 3/6 = 1/2, c₁/c₂ = 5/10 = 1/2. All equal → infinitely many solutions (same line). If c₁/c₂ ≠ 1/2, no solution.

How do you convert a word problem into linear equations?

Steps: (1) Identify the unknown quantities and assign variables (x, y). (2) Write two equations from the given conditions. (3) Solve the system using substitution or elimination. Tip: look for phrases like 'sum is', 'difference is', 'is twice', 'years ago/later', 'fraction' to identify the relationships.

Substitution and Elimination Method

Pair of Linear Equations in Two Variables • Class 10 Mathematics • NCERT • CBSE

A pair of linear equations ax + by + c = 0 and px + qy + r = 0 can be solved by graphical, substitution, elimination, or cross-multiplication methods. The substitution method involves expressing one variable in terms of the other and substituting.

Key Formulas

Intersecting: a₁/a₂ ≠ b₁/b₂ → unique solution
Coincident: a₁/a₂ = b₁/b₂ = c₁/c₂ → infinite solutions
Parallel: a₁/a₂ = b₁/b₂ ≠ c₁/c₂ → no solution
Cross-multiplication: x/(b₁c₂−b₂c₁) = y/(c₁a₂−c₂a₁) = 1/(a₁b₂−a₂b₁)

Frequently Asked Questions

What are the methods to solve a pair of linear equations?: Four methods: (1) Graphical — plot both lines, intersection is solution; (2) Substitution — express x or y from one equation, substitute in other; (3) Elimination — multiply equations to equalise one coefficient, then add/subtract to eliminate it; (4) Cross-multiplication — direct formula using coefficients. Elimination and substitution are most common in CBSE exams.
How do you check if a pair of linear equations has no solution?: Compare ratios of coefficients: a₁/a₂ = b₁/b₂ ≠ c₁/c₂ → no solution (parallel lines). Example: 2x + 3y = 5 and 4x + 6y = 10: a₁/a₂ = 2/4 = 1/2, b₁/b₂ = 3/6 = 1/2, c₁/c₂ = 5/10 = 1/2. All equal → infinitely many solutions (same line). If c₁/c₂ ≠ 1/2, no solution.
How do you convert a word problem into linear equations?: Steps: (1) Identify the unknown quantities and assign variables (x, y). (2) Write two equations from the given conditions. (3) Solve the system using substitution or elimination. Tip: look for phrases like 'sum is', 'difference is', 'is twice', 'years ago/later', 'fraction' to identify the relationships.

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