Math Problem Statement
Solve the equation for x, using the method for evaluating determinants: | 2 -1 -4 | | 1 3 6 | | x 4 3 | = -47
Solution
The equation shown in the image is a system of linear equations represented in a 3x3 matrix form. The system is:
2 & -1 & -4 \\ 1 & 3 & 6 \\ x & 4 & 3 \end{vmatrix} = -47$$ We are tasked with solving for $$x$$ using determinants. To solve this, we'll compute the determinant of the 3x3 matrix, set it equal to $$-47$$, and solve for $$x$$. Let me compute the determinant of the matrix and solve for $$x$$.The solution for $$x$$ is: $$x = -\frac{2}{3}$$ Would you like further details on the steps used to compute this determinant? Here are 5 related questions to help expand on this concept: 1. How is the determinant of a 3x3 matrix calculated using cofactor expansion? 2. What does it mean for a system of linear equations when the determinant of the coefficient matrix is non-zero? 3. How can Cramer's rule be applied to solve systems of equations using determinants? 4. What are the implications of a zero determinant in a system of equations? 5. How can matrix operations be used to solve systems with more than three variables? **Tip**: When solving for a variable using a determinant, ensure that the matrix remains square (same number of rows and columns).Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Determinants
Systems of Linear Equations
Formulas
Determinant of a 3x3 matrix formula: det(A) = a(ei − fh) − b(di − fg) + c(dh − eg)
Theorems
Cramer's Rule
Cofactor Expansion
Suitable Grade Level
College-level (Introductory Linear Algebra or Advanced High School Math)