Math Problem Statement
Solve using Cramer's Rule: 2x1 - x2 = 12, 3x1 + x2 = 7.
Solution
To solve this system of linear equations using Cramer’s Rule, let’s start by identifying the system:
2x_1 - x_2 = 12 \\ 3x_1 + x_2 = 7 \end{cases}$$ ### Step 1: Set Up the Determinant of the Coefficient Matrix The coefficient matrix $$A$$ and its determinant $$\det(A)$$ are as follows: $$A = \begin{bmatrix} 2 & -1 \\ 3 & 1 \end{bmatrix}$$ $$\det(A) = (2)(1) - (-1)(3) = 2 + 3 = 5$$ ### Step 2: Determine $$x_1$$ and $$x_2$$ Using Cramer’s Rule According to Cramer’s Rule, each variable $$x_1$$ and $$x_2$$ can be found by substituting the constants from the right side of the equations into the respective columns of the coefficient matrix. #### Finding $$x_1$$ To find $$x_1$$, replace the first column of $$A$$ with the constants from the right side of the equations: $$A_{x_1} = \begin{bmatrix} 12 & -1 \\ 7 & 1 \end{bmatrix}$$ $$\det(A_{x_1}) = (12)(1) - (-1)(7) = 12 + 7 = 19$$ Then, using Cramer’s Rule: $$x_1 = \frac{\det(A_{x_1})}{\det(A)} = \frac{19}{5}$$ #### Finding $$x_2$$ To find $$x_2$$, replace the second column of $$A$$ with the constants: $$A_{x_2} = \begin{bmatrix} 2 & 12 \\ 3 & 7 \end{bmatrix}$$ $$\det(A_{x_2}) = (2)(7) - (12)(3) = 14 - 36 = -22$$ Then, using Cramer’s Rule: $$x_2 = \frac{\det(A_{x_2})}{\det(A)} = \frac{-22}{5}$$ ### Solution The solution to the system is: $$x_1 = \frac{19}{5}, \quad x_2 = -\frac{22}{5}$$ Would you like more details on Cramer's Rule or any part of this solution? --- Here are five related questions for further exploration: 1. How can Cramer's Rule be applied to a 3x3 system of equations? 2. What are the conditions under which Cramer's Rule is not applicable? 3. How does Cramer’s Rule connect to the concept of matrix inversion? 4. What are some practical applications of Cramer’s Rule in real-world problems? 5. How does the determinant influence the solvability of a system? **Tip:** Remember that Cramer’s Rule is efficient for small systems but becomes computationally expensive for larger systems due to the determinant calculations.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Linear Equations
Determinants
Formulas
Determinant formula for a 2x2 matrix: det(A) = ad - bc
Cramer's Rule formulas: x1 = det(A_x1) / det(A), x2 = det(A_x2) / det(A)
Theorems
Cramer's Rule
Suitable Grade Level
Grades 10-12
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