Question

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

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asked 2021-03-03
Find values of a and b such that the system of linear equations has no solution.
x+2y=3
ax+by=-9

Answers (1)

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.
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