Determine the number of solutions to this system of linear equations: 2x+y=6 4x+2y=8 (1 point)
The given system of linear equations can be rewritten in matrix form as:
\[\begin{pmatrix}2 & 1 \\ 4 & 2\end{pmatrix}\begin{pmatrix}x \\ y\end{pmatrix}=\begin{pmatrix}6 \\ 8\end{pmatrix}\]
The coefficient matrix is:
\[A=\begin{pmatrix}2 & 1 \\ 4 & 2\end{pmatrix}\]
The augmented matrix is:
\[\begin{pmatrix}2 & 1 & | & 6 \\ 4 & 2 & | & 8\end{pmatrix}\]
We can perform row operations to put the augmented matrix in row-echelon form:
\[\begin{pmatrix}1 & \frac{1}{2} & | & 3 \\ 0 & 0 & | & 0\end{pmatrix}\]
The augmented matrix has a row of all zeros, which correspond to the equation \(0x+0y=0\) or simply \(0=0\). This equation is always true, so there are infinitely many solutions to the system of linear equations.
Therefore, the number of solutions to the given system of linear equations is infinite.