MP Board · Class 10 · Mathematics · Pair of Linear Equations in Two VariablesExplain the conditions for the consistency and inconsistency of a pair of linear equations in two variables: $$a1x + b1y + c1 = 0$$ $$a2x + b2y + c2 = 0$$\nDiscuss all three cases based on the ratios of their coefficients algebraically and graphically, providing a suitable example for each case.

Consistency and Inconsistency of a Pair of Linear Equations

\nA pair of linear equations in two variables represented in general form as:

1. $a_1x + b_1y + c_1 = 0$
2. $a_2x + b_2y + c_2 = 0$ \ncan be analyzed by comparing the ratios of their coefficients $\frac{a_1}{a_2}$, $\frac{b_1}{b_2}$, and $\frac{c_1}{c_2}$.

Case 1: Intersecting Lines (Unique Solution)

• Algebraic Condition: $\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$
• Graphical Interpretation: The two lines represented by the equations intersect at a single unique point.
• Consistency: The system of equations is Consistent because it has at least one solution.
• Example: $$x - 2y = 0 \quad \text{and} \quad 3x + 4y - 20 = 0$$ Here, $\frac{a_1}{a_2} = \frac{1}{3}$ and $\frac{b_1}{b_2} = \frac{-2}{4} = -\frac{1}{2}$. Since $\frac{1}{3} \neq -\frac{1}{2}$, the system has a unique solution $(x = 4, y = 2)$.

Case 2: Coincident Lines (Infinitely Many Solutions)

• Algebraic Condition: $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$
• Graphical Interpretation: The two lines completely overlap each other and are coincident.
• Consistency: The system is Dependent and Consistent as there are infinitely many common points.
• Example: $$2x + 3y - 9 = 0 \quad \text{and} \quad 4x + 6y - 18 = 0$$ Here, $\frac{a_1}{a_2} = \frac{2}{4} = \frac{1}{2}$, $\frac{b_1}{b_2} = \frac{3}{6} = \frac{1}{2}$, and $\frac{c_1}{c_2} = \frac{-9}{-18} = \frac{1}{2}$. Since all ratios are equal, there are infinitely many solutions.

Case 3: Parallel Lines (No Solution)

• Algebraic Condition: $\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$
• Graphical Interpretation: The two lines run parallel to each other and never intersect at any point.
• Consistency: The system is Inconsistent because no common point exists.
• Example: $$x + 2y - 4 = 0 \quad \text{and} \quad 2x + 4y - 12 = 0$$ Here, $\frac{a_1}{a_2} = \frac{1}{2}$, $\frac{b_1}{b_2} = \frac{2}{4} = \frac{1}{2}$, and $\frac{c_1}{c_2} = \frac{-4}{-12} = \frac{1}{3}$. Since $\frac{1}{2} = \frac{1}{2} \neq \frac{1}{3}$, the lines are parallel and there is no solution.

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