Determine the number of solutions of this system of linear equations:

Question

X+2y=-4
4x+8y=-16

(This question requires you to show work)

Answer 1

To determine the number of solutions of this system of linear equations, we can put the equations into matrix form and find the determinant of the coefficient matrix.

The system of equations can be written as:

[[1 2] [[x] [[-4]
[4 8]] * [y]] = [16]]

The determinant of the coefficient matrix is:
det([[1 2]
[4 8]]) = (1*8) - (2*4) = 8 - 8 = 0

Since the determinant is equal to 0, the system of equations has either no solutions or infinite solutions. To determine which, we can reduce the system to row-echelon form.

R2 = R2 - 4*R1
[[1 2]
[0 0]] [[x] [[-4]
[y]] = [-32]]

The row-echelon form gives us the equation:
x + 2y = -4

Since this reduced system has the same number of equations as variables, and they are not conflicting (i.e., no contradiction), the system has infinitely many solutions.