Question

Determine the number of solutions of this system of linear equations: −x+y=3 −2x+2y=3 This question requires you to show your work. (1 point) Responses No Solutions No Solutions One Solution One Solution Infinitely Many Solutions

Answer 1

To determine the number of solutions of this system of linear equations, we can rewrite the equations in the form Ax = B, where A is the matrix of coefficients and B is the matrix of constants.

The system of equations can be represented as:

A = \begin{bmatrix} -1 & 1 \\ -2 & 2 \end{bmatrix}
B = \begin{bmatrix} 3 \\ 3 \end{bmatrix}

To find the determinant of matrix A, we calculate:
det(A) = (-1)(2) - (-2)(1) = -2 + 2 = 0

Since the determinant of matrix A is 0, the system of equations has either no solution, or infinitely many solutions. To determine which is the case, we can reduce the matrix A to its reduced row-echelon form:

\begin{bmatrix} -1 & 1 & 3 \\ -2 & 2 & 3 \end{bmatrix}

R2 = R2 + 2R1
\begin{bmatrix} -1 & 1 & 3 \\ 0 & 4 & 9 \end{bmatrix}

R2 = (1/4)R2
\begin{bmatrix} -1 & 1 & 3 \\ 0 & 1 & 9/4 \end{bmatrix}

R1 = R1 + R2
\begin{bmatrix} -1 & 0 & 3 + 1 \\ 0 & 1 & 9/4 \end{bmatrix}
\begin{bmatrix} -1 & 0 & 4 \\ 0 & 1 & 9/4 \end{bmatrix}

The system of equations has been reduced to:
-x = 4
y = 9/4

This implies that the system of equations is inconsistent, as the equations cannot be simultaneously true. Therefore, the system of equations has no solution.