Math Problem Statement
Write the system of equations as a matrix equation AX = B, with A as the coefficient matrix of the system: x + 5y = 7, -6x + 8y = 34.
Solution
To express this system of equations as a matrix equation , let’s start by identifying the coefficients and constants.
The system given is:
x + 5y = 7 \\ -6x + 8y = 34 \end{cases}$$ ### Step 1: Identify the Coefficient Matrix $$A$$ The coefficient matrix $$A$$ consists of the coefficients of $$x$$ and $$y$$ from each equation: $$A = \begin{bmatrix} 1 & 5 \\ -6 & 8 \end{bmatrix}$$ ### Step 2: Define the Variable Matrix $$X$$ The variable matrix $$X$$ contains the variables $$x$$ and $$y$$: $$X = \begin{bmatrix} x \\ y \end{bmatrix}$$ ### Step 3: Define the Constant Matrix $$B$$ The constant matrix $$B$$ is made up of the constants from the right side of each equation: $$B = \begin{bmatrix} 7 \\ 34 \end{bmatrix}$$ ### Matrix Equation Representation Now we can write the system as a matrix equation $$AX = B$$: $$\begin{bmatrix} 1 & 5 \\ -6 & 8 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 7 \\ 34 \end{bmatrix}$$ This matrix equation represents the original system of equations. Would you like further details on how to solve this matrix equation? --- ### Related Questions to Explore 1. How can we solve the matrix equation $$AX = B$$ using matrix inversion? 2. What methods are available to determine if a system of linear equations is consistent? 3. How does Gaussian elimination work to solve a system of linear equations? 4. How would you interpret the determinant of the coefficient matrix $$A$$ in this context? 5. Can a system of equations have infinitely many solutions, and how would that look in matrix form? ### Tip: When converting a system of equations into a matrix form, always align the variables in the same order in each equation to ensure correct matrix representation.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Representation of Linear Systems
Formulas
Matrix Equation AX = B
Theorems
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Suitable Grade Level
Grades 10-12