Math Problem Statement
Find x, y, z for the given matrix equation: [x, 3; z, 2] + [2 - y, z; 2 - z, -x] = [3, 7; 2, 0]
Solution
The equation in the image involves two 2x2 matrices being added together and set equal to another matrix. The equation is as follows:
x & 3 \\ z & 2 \end{bmatrix} + \begin{bmatrix} 2 - y & z \\ 2 - z & -x \end{bmatrix} = \begin{bmatrix} 3 & 7 \\ 2 & 0 \end{bmatrix}$$ To solve for $$x$$, $$y$$, and $$z$$, we need to add the two matrices on the left-hand side and equate the result to the matrix on the right-hand side. The matrix addition gives: $$\begin{bmatrix} x + (2 - y) & 3 + z \\ z + (2 - z) & 2 + (-x) \end{bmatrix} = \begin{bmatrix} 3 & 7 \\ 2 & 0 \end{bmatrix}$$ Now, let's compare the elements of the two matrices to form a system of equations: 1. $$x + (2 - y) = 3$$ 2. $$3 + z = 7$$ 3. $$z + (2 - z) = 2$$ 4. $$2 - x = 0$$ ### Simplifying each equation: 1. $$x + 2 - y = 3$$ → $$x - y = 1$$ 2. $$3 + z = 7$$ → $$z = 4$$ 3. $$z + 2 - z = 2$$ → This is always true (identity). 4. $$2 - x = 0$$ → $$x = 2$$ Now that $$x = 2$$ and $$z = 4$$, substitute $$x = 2$$ into the first equation: $$2 - y = 1 \quad \Rightarrow \quad y = 1$$ ### Final values: - $$x = 2$$ - $$y = 1$$ - $$z = 4$$ Let me know if you'd like more details or clarifications. Here are 5 related questions for deeper understanding: 1. What is the general process for adding matrices? 2. How can matrix equations be used to solve systems of equations? 3. What other types of problems can be solved using matrix operations? 4. How do you check if matrix solutions are correct? 5. What are some real-world applications of matrix equations? **Tip:** When solving matrix equations, always verify if your solution satisfies the original matrix equation by substituting the values back into the matrices.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Matrix Equations
Systems of Equations
Matrix Addition
Formulas
Matrix addition
Solving linear equations
Theorems
Basic properties of matrix addition
Linear equations
Suitable Grade Level
Grades 10-12
Related Recommendation
Matrix Equation Solution: Finding x, y, z in a Matrix System
Solve Matrix Equation: Find x, y, and z
Matrix Algebra Problem: Find x + z + y for Matrix Equation
Solving a System of Equations via Matrix Multiplication: [x + y, y; 2x, x - y] * [2; -1] = [3; 2]
Solving Matrix Equation to Find x, y, z, and a