Use Cramer's rule to find the value x, y and z in the equations: 2x + 3z = −1, y + 2z = 5 and x + y = 1.

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To solve the system of equations using Cramer's rule, we first write the equations in standard form $ax + by + cz = d$ :

$2x + 0y + 3z = -1$
$0x + 1y + 2z = 5$
$1x + 1y + 0z = 1$

Form the coefficient matrix $A$ and calculate its determinant, $\det(A)$ .
$A = \begin{bmatrix} 2 & 0 & 3 \\ 0 & 1 & 2 \\ 1 & 1 & 0 \end{bmatrix}$ $\det(A) = 2(1 \cdot 0 - 2 \cdot 1) - 0(0 \cdot 0 - 2 \cdot 1) + 3(0 \cdot 1 - 1 \cdot 1)$ $\det(A) = 2(-2) - 0 + 3(-1)$ $\det(A) = -4 - 3 = -7$
Form the matrix $A_x$ by replacing the first column of $A$ with the constant terms and calculate its determinant, $\det(A_x)$ .
$A_x = \begin{bmatrix} -1 & 0 & 3 \\ 5 & 1 & 2 \\ 1 & 1 & 0 \end{bmatrix}$ $\det(A_x) = -1(1 \cdot 0 - 2 \cdot 1) - 0(5 \cdot 0 - 2 \cdot 1) + 3(5 \cdot 1 - 1 \cdot 1)$ $\det(A_x) = -1(-2) - 0 + 3(4)$ $\det(A_x) = 2 + 12 = 14$
Form the matrix $A_y$ by replacing the second column of $A$ with the constant terms and calculate its determinant, $\det(A_y)$ .
$A_y = \begin{bmatrix} 2 & -1 & 3 \\ 0 & 5 & 2 \\ 1 & 1 & 0 \end{bmatrix}$ $\det(A_y) = 2(5 \cdot 0 - 2 \cdot 1) - (-1)(0 \cdot 0 - 2 \cdot 1) + 3(0 \cdot 1 - 5 \cdot 1)$ $\det(A_y) = 2(-2) + 1(-2) + 3(-5)$ $\det(A_y) = -4 - 2 - 15 = -21$
Form the matrix $A_z$ by replacing the third column of $A$ with the constant terms and calculate its determinant, $\det(A_z)$ .
$A_z = \begin{bmatrix} 2 & 0 & -1 \\ 0 & 1 & 5 \\ 1 & 1 & 1 \end{bmatrix}$ $\det(A_z) = 2(1 \cdot 1 - 5 \cdot 1) - 0(0 \cdot 1 - 5 \cdot 1) + (-1)(0 \cdot 1 - 1 \cdot 1)$ $\det(A_z) = 2(-4) - 0 - 1(-1)$ $\det(A_z) = -8 + 1 = -7$
Calculate the values of $x, y,$ and $z$ using Cramer's rule.
$x = \frac{\det(A_x)}{\det(A)} = \frac{14}{-7} = -2$ $y = \frac{\det(A_y)}{\det(A)} = \frac{-21}{-7} = 3$ $z = \frac{\det(A_z)}{\det(A)} = \frac{-7}{-7} = 1$
The values are:
$x = -2, y = 3, z = 1$

Use Cramer's rule to find the value x, y and z in the equations: 2x + 3z = -1, y + 2z = 5 and x + y = 1.

Answer

Step-by-step solution

Form the coefficient matrix $A$ and calculate its determinant, $\det(A)$ .

Form the matrix $A_x$ by replacing the first column of $A$ with the constant terms and calculate its determinant, $\det(A_x)$ .

Form the matrix $A_y$ by replacing the second column of $A$ with the constant terms and calculate its determinant, $\det(A_y)$ .

Form the matrix $A_z$ by replacing the third column of $A$ with the constant terms and calculate its determinant, $\det(A_z)$ .

Calculate the values of $x, y,$ and $z$ using Cramer's rule.

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More Mathematics Questions

Use Cramer's rule to find the value x, y and z in the equations: 2x + 3z = -1, y + 2z = 5 and x + y = 1.

Answer

Step-by-step solution

Form the coefficient matrix AAA and calculate its determinant, det⁡(A)\det(A)det(A).

Form the matrix AxA_xAx​ by replacing the first column of AAA with the constant terms and calculate its determinant, det⁡(Ax)\det(A_x)det(Ax​).

Form the matrix AyA_yAy​ by replacing the second column of AAA with the constant terms and calculate its determinant, det⁡(Ay)\det(A_y)det(Ay​).

Form the matrix AzA_zAz​ by replacing the third column of AAA with the constant terms and calculate its determinant, det⁡(Az)\det(A_z)det(Az​).

Calculate the values of x,y,x, y,x,y, and zzz using Cramer's rule.