Math Problem Statement
solve matrix equation
Solve the matrix equation: (3 -1, 2 1) * X = (5, 5)
Solution
The matrix equation given in the image is:
3 & -1 \\ 2 & 1 \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} = \begin{pmatrix} 5 \\ 5 \end{pmatrix}$$ This is equivalent to the system of equations: 1. $$3x_1 - x_2 = 5$$ 2. $$2x_1 + x_2 = 5$$ ### Step 1: Solve the system of equations First, solve for $$x_2$$ from the second equation: $$x_2 = 5 - 2x_1$$ ### Step 2: Substitute into the first equation Substitute $$x_2 = 5 - 2x_1$$ into the first equation: $$3x_1 - (5 - 2x_1) = 5$$ Simplify: $$3x_1 - 5 + 2x_1 = 5$$ $$5x_1 - 5 = 5$$ $$5x_1 = 10$$ $$x_1 = 2$$ ### Step 3: Find $$x_2$$ Substitute $$x_1 = 2$$ back into the expression for $$x_2$$: $$x_2 = 5 - 2(2) = 5 - 4 = 1$$ ### Final Solution: The solution to the matrix equation is: $$X = \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} = \begin{pmatrix} 2 \\ 1 \end{pmatrix}$$ Let me know if you'd like more details on any step, or if you have any questions. --- Here are five related questions: 1. How do you check if a matrix equation has a unique solution? 2. What is the determinant of the matrix in this problem, and how is it relevant? 3. How would you solve this matrix equation using matrix inversion? 4. Can this system of equations be solved using Gaussian elimination? How? 5. How can this matrix equation be represented graphically? **Tip:** When solving systems of equations, if the determinant of the coefficient matrix is non-zero, the system has a unique solution!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Equations
Matrix Multiplication
Formulas
Ax = B (Matrix Equation)
Solving Systems of Linear Equations
Theorems
Determinant Theorem for Unique Solutions
Substitution Method
Suitable Grade Level
Grades 9-12