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Find the value of $k$ for which the following system of equations has a unique solution:
$x\ +\ 2y\ =\ 3$
$5x\ +\ ky\ +\ 7\ =\ 0$
$x\ +\ 2y\ =\ 3$
$5x\ +\ ky\ +\ 7\ =\ 0$
Given: The system of equation is
$x\ +\ 2y\ =\ 3$ ; $5x\ +\ ky\ +\ 7\ =\ 0$
To do: Find the value of $k$ for which the system of the equation has infinitely many solutions
Solution:
The given system of the equation can be written as:
$x\ +\ 2y\ =\ 3$
$5x\ +\ ky\ +\ 7\ =\ 0$
The given system of equation is in the form
$a_1x+b_1y+c_1=0$
$a_2x+b_2y+c_2=0$
Here, $a_1=1,b_1=2 ,c_1=3 ; a_2=5,b_2=k,c_2=7$
For unique solution we must have:
$\frac{a_1}{a_2}$ not equal to $\frac{b_1}{b_2}$
$\frac{1}{5}$ not equal to $\frac{2}{k}$
$k$ not equal to $10$
So, the given system of equation will have a unique solution for all real values except $k=10$
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