Math Problem Statement
solve using cramers rule
Solution
To solve this system of linear equations using Cramer's Rule, I'll first extract the equations from the image:
Cramer's Rule requires us to compute determinants to solve for and . Here are the steps:
Step 1: Write the system in matrix form
The system can be represented as:
6 & 2 \\ -2 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} -8 \\ -3 \end{bmatrix}$$ Let: $$A = \begin{bmatrix} 6 & 2 \\ -2 & 1 \end{bmatrix}, \quad \mathbf{b} = \begin{bmatrix} -8 \\ -3 \end{bmatrix}$$ ### Step 2: Compute the determinant of $$A$$ ($$\text{det}(A)$$) $$\text{det}(A) = (6)(1) - (2)(-2) = 6 + 4 = 10$$ ### Step 3: Compute determinants for $$x$$ and $$y$$ - Replace the first column of $$A$$ with $$\mathbf{b}$$ to compute $$\text{det}(A_x)$$: $$A_x = \begin{bmatrix} -8 & 2 \\ -3 & 1 \end{bmatrix}$$ $$\text{det}(A_x) = (-8)(1) - (2)(-3) = -8 + 6 = -2$$ - Replace the second column of $$A$$ with $$\mathbf{b}$$ to compute $$\text{det}(A_y)$$: $$A_y = \begin{bmatrix} 6 & -8 \\ -2 & -3 \end{bmatrix}$$ $$\text{det}(A_y) = (6)(-3) - (-8)(-2) = -18 - 16 = -34$$ ### Step 4: Solve for $$x$$ and $$y$$ Using Cramer's Rule: $$x = \frac{\text{det}(A_x)}{\text{det}(A)} = \frac{-2}{10} = -0.2$$ $$y = \frac{\text{det}(A_y)}{\text{det}(A)} = \frac{-34}{10} = -3.4$$ ### Final Solution: $$x = -0.2, \quad y = -3.4$$ Let me know if you have questions or need a detailed breakdown of any step! --- **Follow-up Questions**: 1. How does Cramer's Rule apply to larger systems (e.g., 3x3)? 2. What happens if $$\text{det}(A) = 0$$ in Cramer's Rule? 3. Can Cramer's Rule be used efficiently for very large systems of equations? 4. How do you interpret the determinant geometrically? 5. What are some real-life applications of solving systems of linear equations? **Tip**: Always check the determinant of $$A$$. If it is zero, the system either has no solution or infinitely many solutions!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Determinants
Systems of Linear Equations
Formulas
Cramer's Rule: x = det(Ax)/det(A), y = det(Ay)/det(A)
Determinant formula for 2x2 matrix: det(A) = ad - bc
Theorems
Cramer's Rule
Suitable Grade Level
Grades 10-12
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