Solution
Step 1:
Write the equations
[tex]\begin{gathered} \text{x + y + z = 4} \\ \text{x - 2y + z = 7} \\ x\text{ + 3y + 2z = 4} \end{gathered}[/tex]Step 2:
Write the equations in matrix form
[tex]\begin{bmatrix}{1} & {1} & {1} \\ {1} & {-2} & {1} \\ {1} & {3} & {2}\end{bmatrix}\begin{bmatrix}{x} & {} & {} \\ {y} & {} & {} \\ {z} & {} & {}\end{bmatrix}\text{ = }\begin{bmatrix}{} & {4} & {} \\ {} & {7} & {} \\ {} & {4} & {}\end{bmatrix}[/tex]Step 3
Find the determinant
[tex]\begin{gathered} \text{D = }\begin{bmatrix}{1} & {1} & {1} \\ {1} & {-2} & {1} \\ {1} & {3} & {2}\end{bmatrix} \\ =\text{ 1\lparen-4-3\rparen-1\lparen2-1\rparen+ 1\lparen3 + 2\rparen} \\ =\text{ -7 -1 + 5} \\ =\text{ -3} \end{gathered}[/tex]Step 4:
Find the determinant of x
[tex]\begin{gathered} D_x\text{ = }\begin{bmatrix}{4} & {1} & {1} \\ {7} & {-2} & {1} \\ {4} & {3} & {2}\end{bmatrix} \\ =\text{ 4\lparen-4-3\rparen-1\lparen14-4\rparen+1\lparen21+8\rparen} \\ =\text{ -28 - 10 + 29} \\ =\text{ -9} \end{gathered}[/tex]Step 5:
Find the determinat of y
[tex]\begin{gathered} D_y\text{ = }\begin{bmatrix}{1} & {4} & {1} \\ {1} & {7} & {1} \\ {1} & {4} & {2}\end{bmatrix} \\ =\text{ 1\lparen14-4\rparen - 4\lparen2-1\rparen + 1\lparen4-7\rparen} \\ =\text{ 10 - 4 -3} \\ =\text{ 3} \end{gathered}[/tex]Step 6:
Find the determinant of z
[tex]\begin{gathered} D_z\text{ = }\begin{bmatrix}{1} & {1} & {4} \\ {1} & {-2} & {7} \\ {1} & {3} & {4}\end{bmatrix} \\ =\text{ 1\lparen-8-21\rparen - 1\lparen4 - 7\rparen + 4\lparen3 + 2\rparen} \\ =\text{ -29 + 3 + 20} \\ =\text{ -6} \end{gathered}[/tex][tex]\begin{gathered} x\text{ = }\frac{D_x}{D}\text{ = }\frac{-9}{-3}\text{ = 3} \\ \text{y = }\frac{D_y}{D}\text{ = }\frac{3}{-3}\text{ = -1} \\ \text{z = }\frac{D_z}{D}\text{ = }\frac{-6}{-3}\text{ = 2} \end{gathered}[/tex]Final answer
x = 3 , y = -1 . z = 2
