Question

construct the augmented matrix that corresponds to the following system of equations.\

$$\begin{cases}5x + \\dfrac{2y - z}{3} = 4 \\\\ 3(6z - 5x) + y - 8 = 5 \\\\ x - (9 + z) = 4y\\end{cases}$$

Explanation:

Step1: Simplify first equation

Multiply through by 3 to eliminate denominator:
$$3(5x) + 2y - z = 3(4)$$
$$15x + 2y - z = 12$$

Step2: Simplify second equation

Expand and rearrange terms:
$$18z - 15x + y - 8 = 5$$
$$-15x + y + 18z = 13$$

Step3: Simplify third equation

Expand and rearrange terms:
$$x - 9 - z = 4y$$
$$x - 4y - z = 9$$

Step4: Build augmented matrix

Align coefficients of $x,y,z$ and constants:

$$\begin{bmatrix} 15 & 2 & -1 & \mid & 12 \\ -15 & 1 & 18 & \mid & 13 \\ 1 & -4 & -1 & \mid & 9 \end{bmatrix}$$

Answer:

$$\begin{bmatrix} 15 & 2 & -1 & \mid & 12 \\ -15 & 1 & 18 & \mid & 13 \\ 1 & -4 & -1 & \mid & 9 \end{bmatrix}$$