Determine the number of solutions to this system of linear equations

Question

2x+y=6
4x+2y=8

Answer 1

To determine the number of solutions, we can write the system of equations in matrix form as:

$\begin{bmatrix} 2 & 1 \\ 4 & 2 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 6 \\ 8 \end{bmatrix}$

Now, let's row reduce the augmented matrix:

$\begin{bmatrix} 2 & 1 &|& 6 \\ 4 & 2 &|& 8 \end{bmatrix}$

Divide row 2 by 2:

$\begin{bmatrix} 2 & 1 &|& 6 \\ 2 & 1 &|& 4 \end{bmatrix}$

Subtract 2 times row 1 from row 2:

$\begin{bmatrix} 2 & 1 &|& 6 \\ 0 & 0 &|& -2 \end{bmatrix}$

Since the last row corresponds to an inconsistent equation (0 = -2), there are no solutions to this system of equations.