- The system of equations is \(5x + 6y = 28\) and \(4x + 7y = 29\).
- Linear systems are typically represented by coefficient or augmented matrices.
- The statement refers to an "array" that is not provided in the problem text.
- Because no valid array is presented for comparison, the assertion is false.
- The final answer is \(\boxed{\text{False}}\).
Explanation
- Analyzing the system of equations and the provided assertion
The given system of equations is:
\[ \begin{cases} 5x + 6y = 28 \ 4x + 7y = 29 \end{cases} \]
A system of linear equations can be represented in matrix form as \(AX = B\), where \(A\) is the coefficient matrix, \(X\) is the variable vector, and \(B\) is the constant vector. Alternatively, it can be represented by an augmented matrix \([A|B]\). The provided statement claims that the system "can be represented by the array," but there is no specific array (matrix) provided in the problem description to evaluate. Consequently, without an array presented for comparison, the statement is logically incomplete and cannot be considered true.
- Conclusion on the validity of the statement
Standard matrix representation of the given system would involve a coefficient matrix \(A = \begin{pmatrix} 5 & 6 \ 4 & 7 \end{pmatrix}\) and a column vector of constants \(B = \begin{pmatrix} 28 \ 29 \end{pmatrix}\). An augmented matrix would be expressed as \(\begin{pmatrix} 5 & 6 & 28 \ 4 & 7 & 29 \end{pmatrix}\). Since no such array was included in the prompt, the condition specified in the assertion is not satisfied. Therefore, we conclude that the statement "can be represented by the array" is false.
Examples
Systems of linear equations are fundamental in fields like economics, engineering, and computer science. For example, if a company produces two products, A and B, and has specific constraints on resources (like labor and materials) represented by linear equations, they can use matrix notation to solve for the optimal production levels. Representing these systems correctly as arrays (matrices) is crucial for computer software, such as those using Gaussian elimination or matrix inversion, to solve complex real-world problems efficiently. Understanding how a system relates to its matrix form allows for these automated solutions.