Math Problem Statement
Resuelvelo paso a paso por la regla de Cramer
Resuelvelo paso a paso por la regla de Cramer
Solution
To solve this system of equations using Cramer's rule, we first need to express it in matrix form:
Given equations:
x_1 + x_2 - x_3 = 7 \\ 4x_1 - x_2 + 5x_3 = 4 \\ 6x_1 + x_2 + 3x_3 = 18 \end{cases}$$ This can be represented in matrix form as: $$AX = B$$ where $$A = \begin{pmatrix} 1 & 1 & -1 \\ 4 & -1 & 5 \\ 6 & 1 & 3 \end{pmatrix}, \quad X = \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix}, \quad B = \begin{pmatrix} 7 \\ 4 \\ 18 \end{pmatrix}$$ ### Step 1: Find the Determinant of $$A$$ (denoted as $$\det(A)$$) Using the matrix $$A$$: $$\det(A) = \begin{vmatrix} 1 & 1 & -1 \\ 4 & -1 & 5 \\ 6 & 1 & 3 \end{vmatrix}$$ We expand this determinant along the first row: $$\det(A) = 1 \cdot \begin{vmatrix} -1 & 5 \\ 1 & 3 \end{vmatrix} - 1 \cdot \begin{vmatrix} 4 & 5 \\ 6 & 3 \end{vmatrix} + (-1) \cdot \begin{vmatrix} 4 & -1 \\ 6 & 1 \end{vmatrix}$$ Calculating each minor: 1. $$\begin{vmatrix} -1 & 5 \\ 1 & 3 \end{vmatrix} = (-1)(3) - (5)(1) = -3 - 5 = -8$$ 2. $$\begin{vmatrix} 4 & 5 \\ 6 & 3 \end{vmatrix} = (4)(3) - (5)(6) = 12 - 30 = -18$$ 3. $$\begin{vmatrix} 4 & -1 \\ 6 & 1 \end{vmatrix} = (4)(1) - (-1)(6) = 4 + 6 = 10$$ Substituting these values into $$\det(A)$$: $$\det(A) = 1 \cdot (-8) - 1 \cdot (-18) + (-1) \cdot 10 = -8 + 18 - 10 = 0$$ ### Conclusion Since $$\det(A) = 0$$, Cramer's rule cannot be applied here, as it requires a non-zero determinant for a unique solution. This implies that the system may be either dependent (having infinitely many solutions) or inconsistent (having no solution). Would you like further explanation on alternative methods to solve this system or analyze its consistency? --- ### Related Questions 1. How can we determine if a system is dependent or inconsistent when $$\det(A) = 0$$? 2. What methods can be used to solve this system if Cramer's rule is inapplicable? 3. How do we interpret the determinant of a matrix in terms of the solution of a linear system? 4. What is the significance of a zero determinant in a 3x3 matrix system? 5. Can Gaussian elimination help in solving this system of equations? ### Tip Always check the determinant of the coefficient matrix before applying Cramer's rule, as a zero determinant indicates the rule is not applicable.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Linear Algebra
System of Linear Equations
Determinants
Cramer's Rule
Formulas
Determinant of a 3x3 matrix: det(A) = a(ei − fh) − b(di − fg) + c(dh − eg)
Cramer's Rule: x_i = det(A_i) / det(A)
Theorems
Cramer's Rule
Suitable Grade Level
Grades 11-12