[Math] For what value of k does this system equations have infinite solutions

linear algebrasystems of equations

For what value of $k$ the system

kx+ay=5 \\
ax+ky=k \\

has infinite solutions?

Honestly, i don't know how to start this problem, i saw that this have something to do with matrices and the determinants, but this is meant to be done without that.

The possible answers are:

$A) $ $-5$ or $5$

$B) \sqrt 5$ or $-\sqrt 5$

$C) -25$ or $25$

$D) 0$

$E)$ Can't be determinated

Best Answer

Since both the equations have infinite solutions, they must coincide with each other

This gives:

$\frac{k}{a}$ = $\frac{a}{k}$ = $\frac{5}{k}$

Considering the equation $\frac{a}{k}$ = $\frac{5}{k}$ and solving it, $a$ turns out to be $5$.

Considering the equation $\frac{k}{a}$ = $\frac{a}{k}$ and knowing that $a = 5$ and solving it, $k^2$ turns out to be 25.

This means $k$ can be $5$ or $-5$.

Hence option $A$ is right.

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