Math Problem Statement

All the solution to [1 2 [X = [ 10 .The equation of line is 3 6 ] Y] 20 ]

Solution

The problem involves solving a system of linear equations expressed in matrix form. The equation given is:

1 & 2 \\ 3 & 6 \end{bmatrix} \begin{bmatrix} X \\ Y \end{bmatrix} = \begin{bmatrix} 10 \\ 20 \end{bmatrix}.$$ ### Step 1: Analyze the system The matrix equation represents the system: $$1 \cdot X + 2 \cdot Y = 10$$ $$3 \cdot X + 6 \cdot Y = 20.$$ ### Step 2: Check for consistency The coefficient matrix $$\begin{bmatrix} 1 & 2 \\ 3 & 6 \end{bmatrix}$$ has dependent rows, as the second row is three times the first row ($$3 \times [1, 2] = [3, 6]$$). Therefore, the determinant of the matrix is zero, and the system is singular (not invertible). This implies the system might have either infinitely many solutions or no solution. ### Step 3: Solve for consistency To check for consistency, rewrite the second equation as: $$3(X + 2Y) = 20.$$ Divide through by 3: $$X + 2Y = \frac{20}{3}.$$ Compare this with the first equation, $$X + 2Y = 10$$. Since these are not equal, the system is **inconsistent** and has **no solution**. ### Conclusion: The given system of equations has no solution because the equations contradict each other. --- If you'd like, I can delve deeper into dependent systems, provide alternative approaches to analyze the matrix, or explore related concepts! Here are some expanded questions for exploration: 1. How do you determine if a matrix is singular or invertible? 2. What are the implications of a singular matrix in systems of equations? 3. How do you interpret dependent rows in a matrix geometrically? 4. Can we reformulate this problem with consistent equations to explore infinite solutions? 5. How does matrix rank relate to the solution of systems of equations? **Tip**: Always check the determinant of the coefficient matrix when solving systems of equations—it can quickly indicate if the system is consistent and unique!

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Matrix Equations
Systems of Linear Equations

Formulas

Matrix equation AX = B, where A is the coefficient matrix, X is the variable vector, and B is the constant vector
Determinant of a matrix to check invertibility

Theorems

Properties of Singular Matrices
Consistency and Inconsistency of Linear Systems

Suitable Grade Level

Grades 10-12

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