To find values of a,b such that given system has no solution:

Linear equations with no solution are inconsistent equation or the graph of such equations do not intersect that is these lines are parallel.

Consider the first equation,

x+2y=3

\(\displaystyle\Rightarrow{2}{y}=-{x}+{3}\)

\(\displaystyle\Rightarrow{y}=-{\frac{{{x}}}{{{2}}}}+{3}\)

Above equation has slope \(\displaystyle-{\frac{{{1}}}{{{2}}}}\).

Consider another equation,

ax+by=-9

\(\displaystyle\Rightarrow{b}{y}=-{a}{x}-{9}\)

\(\displaystyle\Rightarrow{y}=-{\frac{{{a}{x}}}{{{b}}}}-{\frac{{{9}}}{{{b}}}}\)

Slope of above equation is \(\displaystyle-{\frac{{{a}}}{{{b}}}}\).

As given system of equations have no solution the given equations are parallel lines and they have same slopes.

That is,

\(\displaystyle-{\frac{{{a}}}{{{b}}}}=-{\frac{{{1}}}{{{2}}}}\)

\(\displaystyle{R}{i}{g}\leftrightarrow{o}{w}{a}={1}\), b=2

Thus, for a = 1 and b = 2 , the given system has no solution.

Linear equations with no solution are inconsistent equation or the graph of such equations do not intersect that is these lines are parallel.

Consider the first equation,

x+2y=3

\(\displaystyle\Rightarrow{2}{y}=-{x}+{3}\)

\(\displaystyle\Rightarrow{y}=-{\frac{{{x}}}{{{2}}}}+{3}\)

Above equation has slope \(\displaystyle-{\frac{{{1}}}{{{2}}}}\).

Consider another equation,

ax+by=-9

\(\displaystyle\Rightarrow{b}{y}=-{a}{x}-{9}\)

\(\displaystyle\Rightarrow{y}=-{\frac{{{a}{x}}}{{{b}}}}-{\frac{{{9}}}{{{b}}}}\)

Slope of above equation is \(\displaystyle-{\frac{{{a}}}{{{b}}}}\).

As given system of equations have no solution the given equations are parallel lines and they have same slopes.

That is,

\(\displaystyle-{\frac{{{a}}}{{{b}}}}=-{\frac{{{1}}}{{{2}}}}\)

\(\displaystyle{R}{i}{g}\leftrightarrow{o}{w}{a}={1}\), b=2

Thus, for a = 1 and b = 2 , the given system has no solution.