Use cramer's rule to solve the following system of linear equations : 5x-2y+3z=-1, 3x+y-2z=25, 2x-4y 5z=-29

We will use **Cramer's Rule** to solve the system of linear equations. Cramer's Rule applies to systems of linear equations of the form:

\[
AX = B
\]

Where \(A\) is the coefficient matrix, \(X\) is the column matrix of variables, and \(B\) is the column matrix of constants.

The system of equations is:

\[
5x - 2y + 3z = -1 \tag{1}
\]
\[
3x + y - 2z = 25 \tag{2}
\]
\[
2x - 4y + 5z = -29 \tag{3}
\]

### Step 1: Write the coefficient matrix \(A\), the variable matrix \(X\), and the constants matrix \(B\).

1. Coefficient matrix \(A\):
\[
A = \begin{bmatrix}
5 & -2 & 3 \\
3 & 1 & -2 \\
2 & -4 & 5
\end{bmatrix}
\]

2. Variable matrix \(X\):
\[
X = \begin{bmatrix}
x \\
y \\
z
\end{bmatrix}
\]

3. Constants matrix \(B\):
\[
B = \begin{bmatrix}
-1 \\
25 \\
-29
\end{bmatrix}
\]

### Step 2: Calculate the determinant of matrix \(A\), denoted as \( \text{det}(A) \).

The determinant of \(A\) is calculated using:

\[
\text{det}(A) = \begin{vmatrix} 5 & -2 & 3 \\ 3 & 1 & -2 \\ 2 & -4 & 5 \end{vmatrix}
\]

I'll calculate this determinant.The determinant of the coefficient matrix \(A\) is:

\[
\text{det}(A) = -19
\]

Since the determinant is non-zero, we can proceed using **Cramer's Rule** to find the values of \(x\), \(y\), and \(z\).

### Step 3: Apply Cramer's Rule

To find each variable, we will calculate the determinants of matrices formed by replacing columns of \(A\) with the constants matrix \(B\).

- \(x = \frac{\text{det}(A_x)}{\text{det}(A)}\)
- \(y = \frac{\text{det}(A_y)}{\text{det}(A)}\)
- \(z = \frac{\text{det}(A_z)}{\text{det}(A)}\)

Where \(A_x\), \(A_y\), and \(A_z\) are matrices obtained by replacing the first, second, and third columns of \(A\) with the constants matrix \(B\), respectively.

#### Step 3.1: Calculate \( \text{det}(A_x) \)

The matrix \(A_x\) is formed by replacing the first column of \(A\) with the constants matrix \(B\):

\[
A_x = \begin{bmatrix}
-1 & -2 & 3 \\
25 & 1 & -2 \\
-29 & -4 & 5
\end{bmatrix}
\]

I'll calculate \( \text{det}(A_x) \).The determinant of the matrix \(A_x\) is:

\[
\text{det}(A_x) = -76
\]

#### Step 3.2: Calculate \( \text{det}(A_y) \)

The matrix \(A_y\) is formed by replacing the second column of \(A\) with the constants matrix \(B\):

\[
A_y = \begin{bmatrix}
5 & -1 & 3 \\
3 & 25 & -2 \\
2 & -29 & 5
\end{bmatrix}
\]

I'll calculate \( \text{det}(A_y) \).The determinant of the matrix \(A_y\) is:

\[
\text{det}(A_y) = -57
\]

#### Step 3.3: Calculate \( \text{det}(A_z) \)

The matrix \(A_z\) is formed by replacing the third column of \(A\) with the constants matrix \(B\):

\[
A_z = \begin{bmatrix}
5 & -2 & -1 \\
3 & 1 & 25 \\
2 & -4 & -29
\end{bmatrix}
\]

I'll calculate \( \text{det}(A_z) \).It seems like I can’t do more advanced data analysis right now. Please try again later.

However, we can continue solving this manually or discuss anything else related to Cramer's rule.

Would you like me to guide you through the steps manually, or do you have any questions so far?

Here are some related questions to explore further:

1. How can we apply Cramer's rule for a larger system of equations?
2. What happens if the determinant of the coefficient matrix is zero?
3. How does Cramer's rule compare with other methods like Gaussian elimination?
4. Can Cramer's rule be used for non-square systems of equations?
5. What are the limitations of using Cramer's rule in solving systems of linear equations?

**Tip:** Cramer's rule is only efficient for small systems of equations; for larger systems, other methods like matrix factorization are typically faster.

Use cramer's rule to solve the following system of linear equations: 5x-2y+3z=-1, 3x+y-2z=25, 2x-4y+5z=-29

Learn how to solve a system of linear equations using Cramer's Rule with detailed calculations. This problem involves solving the system 5x - 2y + 3z = -1, 3x + y - 2z = 25, and 2x - 4y + 5z = -29 using determinants and matrix algebra. Suitable for advanced high school and early college students.

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