Solve the following system of linear equations using Cramer's rule: x + 2y + 3z = 1, 3x + 2y + z = 4, x + 3y…

Question

Solve the following system of linear equations using Cramer's rule: x + 2y + 3z = 1, 3x + 2y + z = 4, x + 3y + 2z = 0

ScanSolve AI Tutor · Accepted Answer

To solve the system of equations using Cramer's Rule:

$x + 2y + 3z = 1$
$3x + 2y + z = 4$
$x + 3y + 2z = 0$

Write the coefficient matrix $A$ and the constant vector $B$ .
$A = \begin{bmatrix} 1 & 2 & 3 \\ 3 & 2 & 1 \\ 1 & 3 & 2 \end{bmatrix} \quad \text{and} \quad B = \begin{bmatrix} 1 \\ 4 \\ 0 \end{bmatrix}$
Calculate the determinant of the coefficient matrix, $D$ .
$D = \det(A) = \begin{vmatrix} 1 & 2 & 3 \\ 3 & 2 & 1 \\ 1 & 3 & 2 \end{vmatrix}$ $D = 1 \begin{vmatrix} 2 & 1 \\ 3 & 2 \end{vmatrix} - 2 \begin{vmatrix} 3 & 1 \\ 1 & 2 \end{vmatrix} + 3 \begin{vmatrix} 3 & 2 \\ 1 & 3 \end{vmatrix}$ $D = 1((2)(2) - (1)(3)) - 2((3)(2) - (1)(1)) + 3((3)(3) - (2)(1))$ $D = 1(4 - 3) - 2(6 - 1) + 3(9 - 2)$ $D = 1(1) - 2(5) + 3(7)$ $D = 1 - 10 + 21 = 12$
Calculate $D_x$ by replacing the first column of $A$ with $B$ .
$D_x = \begin{vmatrix} 1 & 2 & 3 \\ 4 & 2 & 1 \\ 0 & 3 & 2 \end{vmatrix}$ $D_x = 1 \begin{vmatrix} 2 & 1 \\ 3 & 2 \end{vmatrix} - 2 \begin{vmatrix} 4 & 1 \\ 0 & 2 \end{vmatrix} + 3 \begin{vmatrix} 4 & 2 \\ 0 & 3 \end{vmatrix}$ $D_x = 1((2)(2) - (1)(3)) - 2((4)(2) - (1)(0)) + 3((4)(3) - (2)(0))$ $D_x = 1(4 - 3) - 2(8 - 0) + 3(12 - 0)$ $D_x = 1(1) - 2(8) + 3(12)$ $D_x = 1 - 16 + 36 = 21$
Calculate $D_y$ by replacing the second column of $A$ with $B$ .
$D_y = \begin{vmatrix} 1 & 1 & 3 \\ 3 & 4 & 1 \\ 1 & 0 & 2 \end{vmatrix}$ $D_y = 1 \begin{vmatrix} 4 & 1 \\ 0 & 2 \end{vmatrix} - 1 \begin{vmatrix} 3 & 1 \\ 1 & 2 \end{vmatrix} + 3 \begin{vmatrix} 3 & 4 \\ 1 & 0 \end{vmatrix}$ $D_y = 1((4)(2) - (1)(0)) - 1((3)(2) - (1)(1)) + 3((3)(0) - (4)(1))$ $D_y = 1(8 - 0) - 1(6 - 1) + 3(0 - 4)$ $D_y = 1(8) - 1(5) + 3(-4)$ $D_y = 8 - 5 - 12 = -9$
Calculate $D_z$ by replacing the third column of $A$ with $B$ .
$D_z = \begin{vmatrix} 1 & 2 & 1 \\ 3 & 2 & 4 \\ 1 & 3 & 0 \end{vmatrix}$ $D_z = 1 \begin{vmatrix} 2 & 4 \\ 3 & 0 \end{vmatrix} - 2 \begin{vmatrix} 3 & 4 \\ 1 & 0 \end{vmatrix} + 1 \begin{vmatrix} 3 & 2 \\ 1 & 3 \end{vmatrix}$ $D_z = 1((2)(0) - (4)(3)) - 2((3)(0) - (4)(1)) + 1((3)(3) - (2)(1))$ $D_z = 1(0 - 12) - 2(0 - 4) + 1(9 - 2)$ $D_z = 1(-12) - 2(-4) + 1(7)$ $D_z = -12 + 8 + 7 = 3$
Apply Cramer's Rule to find $x, y, z$.
$x = \frac{D_x}{D} = \frac{21}{12} = \frac{7}{4}$ $y = \frac{D_y}{D} = \frac{-9}{12} = -\frac{3}{4}$ $z = \frac{D_z}{D} = \frac{3}{12} = \frac{1}{4}$
The solution is:
$x = \boxed{\frac{7}{4}}$ $y = \boxed{-\frac{3}{4}}$ $z = \boxed{\frac{1}{4}}$

Solve the following system of linear equations using Cramer's rule: x + 2y + 3z = 1, 3x + 2y + z = 4, x + 3y + 2z = 0

Answer

Step-by-step solution

Write the coefficient matrix $A$ and the constant vector $B$ .

Calculate the determinant of the coefficient matrix, $D$ .

Calculate $D_x$ by replacing the first column of $A$ with $B$ .

Calculate $D_y$ by replacing the second column of $A$ with $B$ .

Calculate $D_z$ by replacing the third column of $A$ with $B$ .

Apply Cramer's Rule to find $x, y, z$.

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More Mathematics Questions

Solve the following system of linear equations using Cramer's rule: x + 2y + 3z = 1, 3x + 2y + z = 4, x + 3y + 2z = 0

Answer

Step-by-step solution

Write the coefficient matrix AAA and the constant vector BBB.

Calculate the determinant of the coefficient matrix, DDD.

Calculate DxD_xDx​ by replacing the first column of AAA with BBB.

Calculate DyD_yDy​ by replacing the second column of AAA with BBB.

Calculate DzD_zDz​ by replacing the third column of AAA with BBB.

Apply Cramer's Rule to find $x, y, z$.

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More Mathematics Questions

Write the coefficient matrix $A$ and the constant vector $B$ .

Calculate the determinant of the coefficient matrix, $D$ .

Calculate $D_x$ by replacing the first column of $A$ with $B$ .

Calculate $D_y$ by replacing the second column of $A$ with $B$ .

Calculate $D_z$ by replacing the third column of $A$ with $B$ .