The pair of equation x + 2y + 5 = 0 and - 3x - 6y + 1 = 0 has

Given, equations are x + 2y + 5 = 0 and - 3x - 6y + 1 = 0.

Comparing the equations with general equation of the form:

ax + by + c = 0;

a₁x + b₁y + c₁ = x + 2y + 5

a₂x + b₂y + c₂ = - 3x - 6y + 1

Here, a₁ = 1, b₁ = 2, c₁ = 5

And a₂ = - 3, b₂ = - 6, c₂ = 1

Taking the ratio of coefficients to compare -

a₁ /a₂ = - 1/3

b₁ /b₂ = - 1/3

c₁ /c₂ = 5/1;

here a₁/a₂ = b₁/b₂ c₁/c₂

This represents pair of parallel lines.

Hence, the pair of equations has no solution.

Alternative solution -

We know for a line y = mx +c;

m represents slope of line.

So, we can write this lines in that form first -

For line: x + 2y + 5 = 0

2y = - 5 - x

Hence slope of first line is - 1/2.

For line: - 3x - 6y + 1 = 0

6y = - 3x + 1

Hence slope of second line is - 1/2.

Slope for both the lines represents the lines are parallel and will never intersect and so will have no solution.