<s>[INST] Why am i getting an incorrect result (0/0)? The given pair of equations are:
$$
3x-2y=0\\
kx+5y=0
$$
Clearly,here, $a_1=3$, $a_2=k$, $b_1=-2$, $b_2=5$, $c_1=0$ and $c_2=0$.
Now, here, I have to find a value of $k$ for which the given system of equations has infinite solutions.
Hence, $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$
Thus, $\frac{3}{k} = -\frac{2}{5} = \frac{0}{0}$
But, $\frac{0}{0}$ is nonsense!
Why am I getting such a wrong result?
Are $c_1$ and $c_2$ not equal to ZERO?
I will be thankful for help!
 [/INST] First, if an equation has no unique solution,
$Δ = 0$,
$3×5 - (-2)k=0$ 
$2k=15$
Using Gaussian Elimination,
$  \left[\begin{array}{rr|r}3 & -2 & 0 \\k & 5 & 0 \\\end{array}\right]$~
$  \left[\begin{array}{rr|r}7.5 & -5 & 0 \\k & 5 & 0 \\\end{array}\right]$~$  \left[\begin{array}{rr|r}7.5 & -5 & 0 \\k + 7.5 & 0 & 0 \\\end{array}\right]$
Therefore, when there is infinite many solutions, $k+7.5=0$ must be true.
Then you have $k+7.5=0$ , i.e. $k=-7.5$
Let $x=t$, where $t$ is any real number, then the solutions of the equation  are  $(x,y)=$    $(t,\frac{3}{2}t)$.
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