Pair of Linear Equations in Two Variables — Important Questions

Graphical method, Substitution method, Elimination method, Cross multiplication

Linear equations can be solved using: 1) Graphical method (plotting lines and finding intersection), 2) Substitution method (expressing one variable in terms of another), 3) Elimination method (eliminating one variable by addition/subtraction), and 4) Cross multiplication method. The quadratic formula is used for quadratic equations, not linear equations.

Express one variable in terms of the other

In the substitution method, the first step is to express one variable in terms of the other from one of the equations (usually the simpler one). Then substitute this expression into the other equation to get an equation in one variable, which can be solved.

k = 4

For infinitely many solutions, we need a₁/a₂ = b₁/b₂ = c₁/c₂. Here: a₁/a₂ = 1/2, c₁/c₂ = 3/6 = 1/2. For infinitely many solutions: b₁/b₂ = 2/k = 1/2. Therefore k = 4.

No solution exists, Lines are parallel, a₁/a₂ = b₁/b₂ ≠ c₁/c₂

Inconsistent equations have no solution because the lines are parallel (never intersect). This occurs when a₁/a₂ = b₁/b₂ but c₁/c₂ is different from these ratios. Lines that intersect have unique solutions, and coincident lines have infinite solutions.