Math Problem Statement
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Solve the system of linear equations represented as a matrix: \(\begin{pmatrix} 2 & -3 \\ 1 & 4 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 4 \\ 13 \end{pmatrix}\).
Solution
The given system of equations is represented as:
2 & -3 \\ 1 & 4 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 4 \\ 13 \end{pmatrix}.$$ We solve this system step by step: ### Step 1: Write the system in equation form Expanding the matrix equation, we have: 1. $$2x - 3y = 4$$ 2. $$x + 4y = 13$$ ### Step 2: Solve the system using substitution or elimination From equation (2): $$x = 13 - 4y$$ Substitute $$x = 13 - 4y$$ into equation (1): $$2(13 - 4y) - 3y = 4$$ $$26 - 8y - 3y = 4$$ $$26 - 11y = 4$$ $$11y = 22$$ $$y = 2$$ Substitute $$y = 2$$ back into $$x = 13 - 4y$$: $$x = 13 - 4(2)$$ $$x = 13 - 8$$ $$x = 5$$ ### Solution: $$x = 5, \, y = 2$$ Would you like me to provide additional details or verify the solution? Here are 5 follow-up questions to explore: 1. How can we use the inverse matrix method to solve this system? 2. What are the conditions for a system of linear equations to have a unique solution? 3. What happens if the determinant of the matrix is zero in this scenario? 4. How does substitution compare to elimination in solving linear systems? 5. Can we solve this system graphically, and how would the solution appear? **Tip:** Always check your solution by substituting the values back into the original equations to ensure they satisfy both equations.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Linear Equations
Formulas
Matrix multiplication: Ax = B
Substitution method for solving linear systems
Theorems
Matrix Representation of Linear Systems
Suitable Grade Level
Grades 9-12