# Solve the systems of equations using matrices.4x+5y=83x-4y=3Leave answer in fraction form.4x+y+z=3-x+y=-11+2z2y+2z=-1-x

Question
Matrices

Solve the systems of equations using matrices.
$$4x+5y=8$$
$$3x-4y=3$$
$$4x+y+z=3$$
$$-x+y=-11+2z$$
$$2y+2z=-1-x$$

2021-02-06

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
$$A^{-1}B=\frac{-1}{31}\begin{bmatrix}-4 & -5 \\-3 & 4 \end{bmatrix}$$$$\begin{bmatrix}8 \\3 \end{bmatrix}$$
$$=\frac{-1}{31}\begin{bmatrix}-47 \\-12 \end{bmatrix}$$
$$x=\frac{47}{31}$$
$$y=\frac{12}{31}$$

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