Dive into the world of knowledge and get your queries resolved at IDNLearn.com. Join our community to receive prompt and reliable responses to your questions from knowledgeable professionals.
Sagot :
Certainly! Let's solve the system of equations using Cramer's Rule step-by-step. Our system of equations is:
[tex]\[ \left\{ \begin{array}{l} -2x + 4y - 9z = -32 \\ -3x + 5y - 9z = -30 \\ -x + 2y - 3z = -10 \end{array} \right. \][/tex]
### Step 1: Write down the coefficient matrix [tex]\( A \)[/tex]
The coefficient matrix [tex]\( A \)[/tex] is the matrix of the coefficients of [tex]\( x, y, \)[/tex] and [tex]\( z \)[/tex]:
[tex]\[ A = \begin{pmatrix} -2 & 4 & -9 \\ -3 & 5 & -9 \\ -1 & 2 & -3 \\ \end{pmatrix} \][/tex]
### Step 2: Calculate the determinant [tex]\( D \)[/tex] of matrix [tex]\( A \)[/tex]
To find [tex]\( D \)[/tex]:
[tex]\[ D = \det(A) \][/tex]
[tex]\[ D = \begin{vmatrix} -2 & 4 & -9 \\ -3 & 5 & -9 \\ -1 & 2 & -3 \end{vmatrix} \][/tex]
### Step 3: Write down the matrices [tex]\( D_x, D_y, \)[/tex] and [tex]\( D_z \)[/tex]
These matrices are formed by replacing one column of [tex]\( A \)[/tex] with the constants from the right-hand side of the equations (-32, -30, -10).
Matrix [tex]\( D_x \)[/tex]: replace the [tex]\( x \)[/tex]-column with the constants
[tex]\[ D_x = \begin{pmatrix} -32 & 4 & -9 \\ -30 & 5 & -9 \\ -10 & 2 & -3 \\ \end{pmatrix} \][/tex]
Matrix [tex]\( D_y \)[/tex]: replace the [tex]\( y \)[/tex]-column with the constants
[tex]\[ D_y = \begin{pmatrix} -2 & -32 & -9 \\ -3 & -30 & -9 \\ -1 & -10 & -3 \\ \end{pmatrix} \][/tex]
Matrix [tex]\( D_z \)[/tex]: replace the [tex]\( z \)[/tex]-column with the constants
[tex]\[ D_z = \begin{pmatrix} -2 & 4 & -32 \\ -3 & 5 & -30 \\ -1 & 2 & -10 \\ \end{pmatrix} \][/tex]
### Step 4: Calculate the determinants [tex]\( D_x, D_y, \)[/tex] and [tex]\( D_z \)[/tex]
To find [tex]\( D_x \)[/tex]:
[tex]\[ D_x = \det(D_x) \][/tex]
[tex]\[ D_x = \begin{vmatrix} -32 & 4 & -9 \\ -30 & 5 & -9 \\ -10 & 2 & -3 \end{vmatrix} \][/tex]
Similarly, we calculate:
[tex]\[ D_y = \det(D_y) \][/tex]
[tex]\[ D_y = \begin{vmatrix} -2 & -32 & -9 \\ -3 & -30 & -9 \\ -1 & -10 & -3 \end{vmatrix} \][/tex]
And finally:
[tex]\[ D_z = \det(D_z) \][/tex]
[tex]\[ D_z = \begin{vmatrix} -2 & 4 & -32 \\ -3 & 5 & -30 \\ -1 & 2 & -10 \end{vmatrix} \][/tex]
### Step 5: Solve for [tex]\( x, y, \)[/tex] and [tex]\( z \)[/tex] using Cramer's Rule
Cramer's Rule gives us the solution as:
[tex]\[ x = \frac{D_x}{D} \][/tex]
[tex]\[ y = \frac{D_y}{D} \][/tex]
[tex]\[ z = \frac{D_z}{D} \][/tex]
Substituting the known determinant values:
[tex]\[ D = 3.0000000000000018 \][/tex]
[tex]\[ D_x = -5.999999999999996 \][/tex]
[tex]\[ D_y = 0.0 \][/tex]
[tex]\[ D_z = 12.000000000000005 \][/tex]
Therefore,
[tex]\[ x = \frac{D_x}{D} = \frac{-5.999999999999996}{3.0000000000000018} = -2 \][/tex]
[tex]\[ y = \frac{D_y}{D} = \frac{0.0}{3.0000000000000018} = 0 \][/tex]
[tex]\[ z = \frac{D_z}{D} = \frac{12.000000000000005}{3.0000000000000018} = 4 \][/tex]
Thus, the solution to the system of equations is:
[tex]\[ \boxed{x = -2, \; y = 0, \; z = 4} \][/tex]
[tex]\[ \left\{ \begin{array}{l} -2x + 4y - 9z = -32 \\ -3x + 5y - 9z = -30 \\ -x + 2y - 3z = -10 \end{array} \right. \][/tex]
### Step 1: Write down the coefficient matrix [tex]\( A \)[/tex]
The coefficient matrix [tex]\( A \)[/tex] is the matrix of the coefficients of [tex]\( x, y, \)[/tex] and [tex]\( z \)[/tex]:
[tex]\[ A = \begin{pmatrix} -2 & 4 & -9 \\ -3 & 5 & -9 \\ -1 & 2 & -3 \\ \end{pmatrix} \][/tex]
### Step 2: Calculate the determinant [tex]\( D \)[/tex] of matrix [tex]\( A \)[/tex]
To find [tex]\( D \)[/tex]:
[tex]\[ D = \det(A) \][/tex]
[tex]\[ D = \begin{vmatrix} -2 & 4 & -9 \\ -3 & 5 & -9 \\ -1 & 2 & -3 \end{vmatrix} \][/tex]
### Step 3: Write down the matrices [tex]\( D_x, D_y, \)[/tex] and [tex]\( D_z \)[/tex]
These matrices are formed by replacing one column of [tex]\( A \)[/tex] with the constants from the right-hand side of the equations (-32, -30, -10).
Matrix [tex]\( D_x \)[/tex]: replace the [tex]\( x \)[/tex]-column with the constants
[tex]\[ D_x = \begin{pmatrix} -32 & 4 & -9 \\ -30 & 5 & -9 \\ -10 & 2 & -3 \\ \end{pmatrix} \][/tex]
Matrix [tex]\( D_y \)[/tex]: replace the [tex]\( y \)[/tex]-column with the constants
[tex]\[ D_y = \begin{pmatrix} -2 & -32 & -9 \\ -3 & -30 & -9 \\ -1 & -10 & -3 \\ \end{pmatrix} \][/tex]
Matrix [tex]\( D_z \)[/tex]: replace the [tex]\( z \)[/tex]-column with the constants
[tex]\[ D_z = \begin{pmatrix} -2 & 4 & -32 \\ -3 & 5 & -30 \\ -1 & 2 & -10 \\ \end{pmatrix} \][/tex]
### Step 4: Calculate the determinants [tex]\( D_x, D_y, \)[/tex] and [tex]\( D_z \)[/tex]
To find [tex]\( D_x \)[/tex]:
[tex]\[ D_x = \det(D_x) \][/tex]
[tex]\[ D_x = \begin{vmatrix} -32 & 4 & -9 \\ -30 & 5 & -9 \\ -10 & 2 & -3 \end{vmatrix} \][/tex]
Similarly, we calculate:
[tex]\[ D_y = \det(D_y) \][/tex]
[tex]\[ D_y = \begin{vmatrix} -2 & -32 & -9 \\ -3 & -30 & -9 \\ -1 & -10 & -3 \end{vmatrix} \][/tex]
And finally:
[tex]\[ D_z = \det(D_z) \][/tex]
[tex]\[ D_z = \begin{vmatrix} -2 & 4 & -32 \\ -3 & 5 & -30 \\ -1 & 2 & -10 \end{vmatrix} \][/tex]
### Step 5: Solve for [tex]\( x, y, \)[/tex] and [tex]\( z \)[/tex] using Cramer's Rule
Cramer's Rule gives us the solution as:
[tex]\[ x = \frac{D_x}{D} \][/tex]
[tex]\[ y = \frac{D_y}{D} \][/tex]
[tex]\[ z = \frac{D_z}{D} \][/tex]
Substituting the known determinant values:
[tex]\[ D = 3.0000000000000018 \][/tex]
[tex]\[ D_x = -5.999999999999996 \][/tex]
[tex]\[ D_y = 0.0 \][/tex]
[tex]\[ D_z = 12.000000000000005 \][/tex]
Therefore,
[tex]\[ x = \frac{D_x}{D} = \frac{-5.999999999999996}{3.0000000000000018} = -2 \][/tex]
[tex]\[ y = \frac{D_y}{D} = \frac{0.0}{3.0000000000000018} = 0 \][/tex]
[tex]\[ z = \frac{D_z}{D} = \frac{12.000000000000005}{3.0000000000000018} = 4 \][/tex]
Thus, the solution to the system of equations is:
[tex]\[ \boxed{x = -2, \; y = 0, \; z = 4} \][/tex]
Your participation is crucial to us. Keep sharing your knowledge and experiences. Let's create a learning environment that is both enjoyable and beneficial. For clear and precise answers, choose IDNLearn.com. Thanks for stopping by, and come back soon for more valuable insights.
