CBSE · Class 10 · Mathematics · Pair of Linear Equations in Two VariablesExplain in detail the algebraic conditions for the consistency and inconsistency of a pair of linear equations in two variables. Discuss the graphical representation, geometric behavior, and the number of solutions for each condition using standard general equations.
Step-by-Step Solution
Pair of Linear Equations in Two Variables: Conditions for Consistency and Inconsistency
\nA pair of linear equations in two variables $x$ and $y$ can be represented in its standard general form as:
- $a_1x + b_1y + c_1 = 0$
- $a_2x + b_2y + c_2 = 0$ \nwhere $a_1, b_1, c_1, a_2, b_2, c_2$ are real numbers such that $a_1^2 + b_1^2 \neq 0$ and $a_2^2 + b_2^2 \neq 0$.
Geometric Interpretation and Conditions
\nThe nature of solutions depends on the comparison of the ratios of their coefficients $\frac{a_1}{a_2}$, $\frac{b_1}{b_2}$, and $\frac{c_1}{c_2}$.
1. Intersecting Lines (Consistent System - Unique Solution)
- Algebraic Condition: $$\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$$
- Graphical Interpretation: The lines representing the two equations intersect at a single unique point on the Cartesian plane.
- Number of Solutions: Exactly one unique solution exists corresponding to the coordinates of the point of intersection.
- Consistency: The system is consistent.
2. Coincident Lines (Dependent Consistent System - Infinitely Many Solutions)
- Algebraic Condition: $$\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$$
- Graphical Interpretation: Both equations represent the exact same line in the plane; one line lies entirely over the other.
- Number of Solutions: There are infinitely many solutions, as every point on the line satisfies both equations.
- Consistency: The system is dependent and consistent.
3. Parallel Lines (Inconsistent System - No Solution)
- Algebraic Condition: $$\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$$
- Graphical Interpretation: The lines drawn on the graph paper are parallel to each other and never cross or touch at any point.
- Number of Solutions: There is no solution.
- Consistency: The system is inconsistent.
Summary Table
| Ratio Comparison | Graphical Representation | Algebraic Interpretation | System 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 | Dependent Consistent |
| $\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$ | Parallel lines | No solution | Inconsistent |
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