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;

a_{1}x + b_{1}y + c_{1} = x + 2y + 5

a_{2}x + b_{2}y + c_{2} = - 3x - 6y + 1

Here, a_{1} = 1, b_{1} = 2, c_{1} = 5

And a_{2} = - 3, b_{2} = - 6, c_{2} = 1

Taking the ratio of coefficients to compare -

a_{1} /a_{2} = - 1/3

b_{1} /b_{2} = - 1/3

c_{1} /c_{2} = 5/1;

here a_{1}/a_{2} = b_{1}/b_{2} c_{1}/c_{2}

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.

2