Question

Solve the systems of equations using matrices.
\(4x+5y=8\)
\(3x-4y=3\)
Leave answer in fraction form.
\(4x+y+z=3\)
\(-x+y=-11+2z\)
\(2y+2z=-1-x\)

Answer 1

Step 1
The given equation
\(4x+5y=8\)
\(3x-4y=3\)
In matrix form
\(\begin{bmatrix}4 & 5 \\3 & -4 \end{bmatrix}\begin{bmatrix}x \\y \end{bmatrix}=\begin{bmatrix}8 \\3 \end{bmatrix}\) \(AX=B\)

Step 2

\(X=A^{-1}B A=\begin{bmatrix}4 & 5 \\3 & -4 \end{bmatrix} A^{-1}=\frac{1}{-16-15}\begin{bmatrix}-4 & -5 \\-3 & 4 \end{bmatrix}=\frac{-1}{31}\begin{bmatrix}-4 & -5 \\-3 & 4 \end{bmatrix}\)
\(B=\begin{bmatrix}8 \\3 \end{bmatrix}\)
Step 3
\(A^{-1}B=\frac{-1}{31}\begin{bmatrix}-4 & -5 \\-3 & 4 \end{bmatrix}\)\(\begin{bmatrix}8 \\3 \end{bmatrix}\)
\(=\frac{-1}{31}\begin{bmatrix}-47 \\-12 \end{bmatrix}\)
\(x=\frac{47}{31}\)
\(y=\frac{12}{31}\)