Math Problem Statement
determine the value(s) of such that the matrix is the augmented matrix of a consistent linear system.
Determine the value(s) of h such that the matrix is the augmented matrix of a consistent linear system.
Solution
The matrix you provided is:
2 & 3 & h \\ 4 & 6 & 7 \end{bmatrix}$$ This is an augmented matrix for a system of two linear equations. To determine the value of $$h$$ such that the system is consistent, we need to ensure that the system does not result in a contradiction (like $$0 = c$$, where $$c$$ is a non-zero constant). ### Step 1: Write the augmented matrix as a system of equations The matrix corresponds to the following system of equations: $$2x + 3y = h$$ $$4x + 6y = 7$$ ### Step 2: Perform row reduction or check for proportionality Notice that the second row is a multiple of the first row: $$(4x + 6y) = 2 \times (2x + 3y)$$ Thus, the two equations are proportional. This implies that for the system to be consistent, the right-hand side of the second equation (7) must also be a multiple of the right-hand side of the first equation (h). Therefore, for consistency: $$7 = 2h$$ ### Step 3: Solve for $$h$$ $$h = \frac{7}{2}$$ ### Conclusion: The value of $$h$$ for which the system is consistent is $$h = \frac{7}{2}$$. --- Would you like further details on the row reduction process or any additional clarifications? Here are some related questions: 1. What happens if $$h \neq \frac{7}{2}$$? 2. How can we check if a matrix represents an inconsistent system? 3. What are the general methods for solving augmented matrices? 4. How do row operations affect the solutions of a linear system? 5. What is the difference between consistent and inconsistent systems? **Tip**: When analyzing an augmented matrix for consistency, always check if the rows are linearly dependent, as this gives important clues about the system's consistency.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Systems
Consistent and Inconsistent Systems
Proportionality
Formulas
System of Linear Equations
Row Reduction
Theorems
Proportionality Theorem in Linear Systems
Suitable Grade Level
Grades 10-12
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