Mathematics: Class 10 — Chapter 3: Pair of Linear Equations in Two Variables

For a pair of linear equations: a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0:

• Intersecting Lines: If a₁/a₂ ≠ b₁/b₂, the lines intersect at a single point. There is exactly one unique solution (Consistent).
• Coincident Lines: If a₁/a₂ = b₁/b₂ = c₁/c₂, the lines coincide. There are infinitely many solutions (Dependent/Consistent).
• Parallel Lines: If a₁/a₂ = b₁/b₂ ≠ c₁/c₂, the lines are parallel. There is no solution (Inconsistent).

• Step 1: Express one variable (say y) in terms of the other variable (say x) from any one of the equations.
• Step 2: Substitute this value of y in the other equation to reduce it to a linear equation in one variable x, and solve it.
• Step 3: Substitute the value of x obtained in Step 2 into the equation from Step 1 to get the value of y.

• Step 1: Multiply both equations by suitable non-zero constants to make the coefficients of one variable (either x or y) numerically equal.
• Step 2: Add or subtract one equation from the other so that one variable gets eliminated, leaving an equation in one variable.
• Step 3: Solve this equation to find the value of the remaining variable.
• Step 4: Substitute this value in either of the original equations to find the value of the other variable.