Answer
$a = 2$ or $a = -2$
Work Step by Step
To find when there is a whole line of solutions (i.e., infinitely many solutions) for the system of equations \(ax + 2y = 0\) and \(2x + ay = 0\), we can use the fact that for infinitely many solutions, the determinant of the coefficient matrix should be zero.
The coefficient matrix is:
\[
\begin{pmatrix}
a & 2 \\
2 & a \\
\end{pmatrix}
\]
The determinant of this matrix should be zero:
\[ \text{det} \left( \begin{pmatrix} a & 2 \\ 2 & a \end{pmatrix} \right) = a^2 - 4 = 0 \]
Now, solve for \(a\):
\[ a^2 - 4 = 0 \]
\[ a^2 = 4 \]
\[ a = \pm 2 \]
So, for \(a = 2\) or \(a = -2\), there will be a whole line of solutions.
