Math Problem Statement
Solve the system of equations resulting from the matrix multiplication: [x + y, y; 2x, x - y] * [2; -1] = [3; 2].
Solution
The given equation involves matrix multiplication. Here's the system we have:
x + y & y \\ 2x & x - y \end{bmatrix} \begin{bmatrix} 2 \\ -1 \end{bmatrix} = \begin{bmatrix} 3 \\ 2 \end{bmatrix}$$ To solve this, we perform the matrix multiplication on the left-hand side: ### First row multiplication: $$(x + y)(2) + (y)(-1) = 3$$ Expanding: $$2(x + y) - y = 3$$ Simplifying: $$2x + 2y - y = 3 \quad \Rightarrow \quad 2x + y = 3 \quad \text{(Equation 1)}$$ ### Second row multiplication: $$(2x)(2) + (x - y)(-1) = 2$$ Expanding: $$4x - (x - y) = 2$$ Simplifying: $$4x - x + y = 2 \quad \Rightarrow \quad 3x + y = 2 \quad \text{(Equation 2)}$$ Now, we have the system of equations: 1. $$2x + y = 3$$ 2. $$3x + y = 2$$ Let's solve this system of linear equations. ### Step 1: Subtract Equation 1 from Equation 2 $$(3x + y) - (2x + y) = 2 - 3$$ $$3x + y - 2x - y = -1$$ $$x = -1$$ ### Step 2: Substitute $$x = -1$$ into Equation 1 $$2(-1) + y = 3$$ $$-2 + y = 3 \quad \Rightarrow \quad y = 5$$ Thus, the solution is: $$x = -1, \quad y = 5$$ Let me know if you need further details or have any questions! --- Here are five related questions: 1. How do you check if a system of equations is consistent? 2. What happens if the determinant of a matrix is zero? 3. Can you always solve a system of linear equations using matrix multiplication? 4. How do you interpret matrix multiplication geometrically? 5. What other methods can be used to solve systems of equations? **Tip:** When solving matrix equations, always check if the determinant is non-zero to ensure the system has a unique solution!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Multiplication
Systems of Linear Equations
Algebra
Formulas
Matrix multiplication formula: [a, b; c, d] * [e; f] = [a*e + b*f; c*e + d*f]
Solving systems of equations using substitution or elimination
Theorems
Properties of matrix multiplication
Solving linear systems of equations
Suitable Grade Level
Grades 9-12