Algebraic Consistency Conditions
For a pair of linear equations:
$$a_1x + b_1y + c_1 = 0 \quad \text{and} \quad a_2x + b_2y + c_2 = 0$$
| Ratio Comparison | Graphical Representation | Algebraic Interpretation | Consistency |
|---|---|---|---|
| $\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$ | Intersecting lines | Exactly one unique solution | Consistent |
| $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$ | Coincident lines | Infinitely many solutions | Consistent (Dependent) |
| $\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$ | Parallel lines | No solution | Inconsistent |
Methods of Algebraic Solution
1. Elimination Method:
Multiply one or both equations by suitable non-zero constants so that the coefficients of one variable become equal, then add or subtract equations to eliminate that variable.
2. Substitution Method:
Express one variable in terms of the other from one equation and substitute that expression into the second equation.
