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(C) The given system of linear equations is:$ax + y = 0$ $(i)$$x + (a + 10)y = 0$ $(ii)$For a system of homogeneous linear equations to have at least

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$2$

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(C) The given system of linear equations is:$ax + y = 0$ $(i)$$x + (a + 10)y = 0$ $(ii)$For a system of homogeneous linear equations to have at least two distinct solutions (i.e.,non-trivial solutions),the determinant of the coefficient matrix must be zero.The coefficient matrix is $A = \begin{bmatrix} a & 1 \ 1 & a + 10 \end{bmatrix}$.Setting the determinant to zero:$|A| = a(a + 10) - (1)(1) = 0$$a^2 + 10a - 1 = 0$This is a quadratic equation in $a$. The roots are given by the quadratic formula $a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$:$a = \frac{-10 \pm \sqrt{10^2 - 4(1)(-1)}}{2(1)}$$a = \frac{-10 \pm \sqrt{100 + 4}}{2}$$a = \frac{-10 \pm \sqrt{104}}{2}$Since the discriminant $D = 104 > 0$,there are $2$ distinct real values of $a$ for which the system has non-trivial (infinitely many) solutions.Therefore,the correct option is $C$.

For how many different values of $a$ does the following system of linear equations have at least two distinct solutions?
$ax + y = 0$
$x + (a + 10)y = 0$

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For how many different values of $a$ does the following system of linear equations have at least two distinct solutions?$ax + y = 0$$x + (a + 10)y = 0$