Question

Represent the following pair of linear equations graphically and hence comment on the condition of consistency of this pair : \( x - 5y = 6; 2x - 10y = 12 \)

Hint:

If the second equation is just a multiple of the first (here, multiplied by 2), they will always be coincident lines.

Answer 1

Step 1: Understanding the Concept:
Consistency is determined by the ratios of coefficients. If the ratios of all coefficients (\( a, b, c \)) are equal, the lines are coincident.
Step 2: Detailed Explanation:
For \( x - 5y = 6 \): Points (6, 0), (1, -1), (11, 1).
For \( 2x - 10y = 12 \): Points (6, 0), (1, -1), (11, 1).
When plotted on a graph, both equations represent the exact same straight line.
Ratio analysis:
\( \frac{a_1}{a_2} = \frac{1}{2} \)
\( \frac{b_1}{b_2} = \frac{-5}{-10} = \frac{1}{2} \)
\( \frac{c_1}{c_2} = \frac{6}{12} = \frac{1}{2} \)
Since \( \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \), the lines are coincident.
Step 3: Final Answer:
The pair is consistent and dependent, having infinitely many solutions.