Question

Find the condition that the system of equations ax+by= c, lx+my= n has a unique solution?

Answer 1

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Answer 2

do you know about matrices?

Answer 3

\[\left[\begin{matrix}a & b \\ l & m\end{matrix}\right] \left[\begin{matrix}x \\ y\end{matrix}\right]=\left[\begin{matrix}c \\ n\end{matrix}\right] \]

Answer 4

to solve you multiply both sides by the inverse of the matrix
you get
\[ \left[\begin{matrix}x \\ y\end{matrix}\right]= \frac{1}{am-bl}\left[\begin{matrix}m & -b \\ -l & a\end{matrix}\right]\left[\begin{matrix}c \\ n\end{matrix}\right] \]

and that has a solution as long as (am-bl) does not equal 0
in other words, the condition is
\[ am \ne bl \]

Answer 5

@phi Nope, I do not know

Answer 6

remember in your previous problem where you wrote the coefficients of x,y
<a,b>
and
<l,m> in this case

if there is a multiplier, (call it "g") so that
ga = l and gb= m (in your problem we found ½ worked)
then that makes the two lines parallel

if we have ga = l then g= l/a
and similarly gb= m means g= m/b

so if g exists, it requires l/a = m/b
or cross-multiplying, bl = am
and fog g to not exist, we want \(bl \ne am\)

Answer 7

*and for g not to exist, we want \( bl \ne am\)