Math Problem Statement
Use Cramer's rule and the calculator provided to find the value of z that satisfies the system of linear equations.
Solution
This image shows the process of using Cramer's Rule to solve a system of linear equations for . The given system of equations is:
-2x + y - 5z &= -2 \\ -y + 2z &= 1 \\ 3x + 5y - 2z &= -4 \end{aligned}$$ To solve for $$z$$ using Cramer's Rule, the determinant of the coefficient matrix $$D$$ is calculated, and the determinant of the modified matrix $$D_z$$ is used to find $$z$$. ### Steps: 1. **Write the Coefficient Matrix**: $$A = \begin{pmatrix} -2 & 1 & -5 \\ 0 & -1 & 2 \\ 3 & 5 & -2 \end{pmatrix}$$ 2. **Calculate the Determinant of the Coefficient Matrix $$D$$**: The image shows that the determinant $$D$$ is already calculated as $$7$$. 3. **Form the Matrix $$A_z$$ (replace the $$z$$-column with constants from the right side of the equations)**: $$A_z = \begin{pmatrix} -2 & 1 & -2 \\ 0 & -1 & 1 \\ 3 & 5 & -4 \end{pmatrix}$$ 4. **Calculate the Determinant of $$A_z$$**: The image indicates that $$\text{det}(A_z) = -1$$. 5. **Apply Cramer's Rule for $$z$$**: $$z = \frac{\text{det}(A_z)}{\text{det}(A)} = \frac{-1}{7} = -\frac{1}{7}$$ Thus, the solution for $$z$$ is $$z = -\frac{1}{7}$$. Would you like further details on this solution or any clarification? Here are 5 related questions to deepen understanding: 1. How is Cramer's Rule used for systems larger than $$3 \times 3$$? 2. What are the conditions under which Cramer's Rule can be applied? 3. Can Cramer's Rule be used if the determinant of the coefficient matrix is zero? Why or why not? 4. How do you find the determinant of a $$3 \times 3$$ matrix manually? 5. How does Cramer's Rule compare to other methods like Gaussian elimination? **Tip**: When applying Cramer's Rule, always check the determinant of the coefficient matrix. If it's zero, the system has either no solution or infinitely many solutions.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Linear Algebra
Cramer's Rule
Determinants
Formulas
Cramer's Rule: z = det(A_z) / det(A)
Determinant of a 3x3 matrix
Theorems
Cramer's Rule
Determinants of matrices
Suitable Grade Level
Grades 10-12