Question

# Find values of a and b such that the system of linear equations has no solution. x+2y=3 ax+by=-9

Equations
Find values of a and b such that the system of linear equations has no solution.
x+2y=3
ax+by=-9

2021-03-05
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.