Math Problem Statement
Find x, y, z, and w.
x 9 (2x − 1)
y 7 4y
(2x − 5) z 9
5 (w + 1) (3y + 5)
Solution
We are given the following matrix equation involving variables , , , and :
x & 9 & (2x - 1) \\ y & 7 & 4y \\ (2x - 5) & z & 9 \\ 5 & (w + 1) & (3y + 5) \end{pmatrix} = \begin{pmatrix} x & 9 & (2x - 1) \\ y & 7 & 4y \\ (2x - 5) & z & 9 \\ 5 & (w + 1) & (3y + 5) \end{pmatrix}$$ ### Solution Process: This is a matrix equality problem. In matrix equations, corresponding entries from both sides of the equation must be equal. So, we can match elements from the matrices and create the system of equations to solve for $$x$$, $$y$$, $$z$$, and $$w$$. ### Step 1: Compare elements from the first row $$x = x,\quad 9 = 9,\quad (2x - 1) = (2x - 1)$$ No new information here. ### Step 2: Compare elements from the second row $$y = y,\quad 7 = 7,\quad 4y = 4y$$ No new information here either. ### Step 3: Compare elements from the third row $$(2x - 5) = (2x - 5),\quad z = z,\quad 9 = 9$$ Again, no new information. ### Step 4: Compare elements from the fourth row $$5 = 5,\quad (w + 1) = (w + 1),\quad (3y + 5) = (3y + 5)$$ No new equations to solve. Thus, this system does not give us any new information or contradictions; it is trivially satisfied for all values of $$x$$, $$y$$, $$z$$, and $$w$$. ### Final Answer: The variables $$x$$, $$y$$, $$z$$, and $$w$$ can be any real numbers. --- Do you need further clarification or details? Let me know if you'd like to explore any specific aspect of this solution. Here are 5 related questions for practice: 1. How do we solve systems of linear equations using matrices? 2. What does it mean when two matrices are equal? 3. How can we use matrix operations to solve systems of equations? 4. What are the conditions for matrix equality in general? 5. Can matrices of different dimensions ever be equal? **Tip:** When solving systems using matrices, ensure that corresponding elements in equal matrices are compared properly for consistency.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Equality
Systems of Equations
Linear Algebra
Formulas
Matrix element equality
Linear systems comparison
Theorems
Matrix equality theorem
Suitable Grade Level
Grades 10-12