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 When the pair of linear equations $kx+4y=5$,$3x+2y=5$ is consistent only?
Given: Pair of linear equations: $kx+4y=5$,$3x+2y=5$
To do: To find the value of $k$ when given pair of equations is consistent only.
Solution:
A pair of linear equations is consistent if it has a solution either a unique or
infinitely many.
Given equations are:
$kx+4y=5 .....( 1)$
$3x+2y=5 ......( 2)$
Here, $a_1=k,\ b_1=4,\ c_1=−5$ and $a_2=3,\ b_2=2,\ c_2=−5$
For given system,
$\frac{a_1}{a_2}=\frac{k}{3}, \frac{b_1}{b_2}=\frac{4}{2}=2$ and $\frac{c_1}{c_2}=\frac{-5}{-5}=1$
Here $\frac{b_1}{b_2}\
eq \frac{c_1}{c_2}$
eq \frac{c_1}{c_2}$
Hence, the system does not have infinitely many solution. So, the equations must have a unique solution.
$\Rightarrow \frac{a_1}{a_2}\
eq\frac{b_1}{b_2}$
eq\frac{b_1}{b_2}$
$\Rightarrow \frac{k}{3}\
eq2$
eq2$
$\Rightarrow k\
eq6$
 eq6$
Thus, for $k\
eq6$, given pair of linear equations is consistent only.
Updated on: 10-Oct-2022
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