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: Write Both Equations
Equation (1): x − 5y = 6
Equation (2): 2x − 10y = 12

Step 2: Compare Ratios of Coefficients
For equation ax + by = c,
a₁ = 1, b₁ = −5, c₁ = 6
a₂ = 2, b₂ = −10, c₂ = 12

Now compute ratios:
a₁/a₂ = 1/2
b₁/b₂ = (−5)/(−10) = 1/2
c₁/c₂ = 6/12 = 1/2

Since
a₁/a₂ = b₁/b₂ = c₁/c₂

Step 3: Interpretation
Both equations represent the same straight line.
So the lines are coincident.

Final Answer:
The pair of equations is consistent and dependent.
They have infinitely many solutions.