Math Problem Statement
Determine values of the variables that will make the following equation true, if possible. If not, state 'Not Possible':
Matrix equation:
[[3, -2w, -2x], [-4, -w, 5x], [-2y, z, 3z]] = [[w + 24, -182], [-y, z], [18, 5]].
Solution
The image contains a matrix equation. I'll transcribe it, solve it, and provide a detailed explanation.
Matrix equation:
3 & -2w & -2x \\ -4 & -w & 5x \\ -2y & z & 3z \end{bmatrix} = \begin{bmatrix} w + 24 & -182 \\ - y & z \\ 18 & 5 \end{bmatrix}$$ We are tasked with finding $$w, x, y, z$$, or determining if it's not possible to solve. --- ### Step-by-step process: 1. **Equating elements of the two matrices:** - **First row:** $$3 = w + 24 \quad \text{(1)}, \quad -2w = -182 \quad \text{(2)}, \quad -2x = \text{unspecified (check the problem!)}.$$ - **Second row:** $$-4 = -y \quad \text{(3)}, \quad -w = z \quad \text{(4)}, \quad 5x = \text{(unspecified)}.$$ - **Third row:** $$-2y = 18 \quad \text{(5)}, \quad z = z \quad \text{(trivial equality)}, \quad 3z = 5 \quad \text{(6)}.$$ 2. **Solving equations:** - From (1): $$w = 3 - 24 = -21.$$ - From (2): $$-2w = -182 \implies w = 91 \quad \text{(conflict with (1)—needs clarification!)}.$$ - From (3): $$-4 = -y \implies y = 4.$$ - From (4): $$-w = z \implies z = -(-21) = 21.$$ - From (5): $$-2y = 18 \implies y = -9 \quad \text{(conflict with (3)—needs clarification!)}.$$ - From (6): $$3z = 5 \implies z = \frac{5}{3}.$$ --- ### Observation: There are multiple conflicts between the equations (e.g., $$w = -21$$ versus $$w = 91$$, $$y = 4$$ versus $$y = -9$$). This suggests that the matrix equation may not be solvable consistently. --- Would you like me to clarify specific portions or check for errors in transcription? --- ### Questions to deepen understanding: 1. Could you clarify or provide any missing values in the matrix? 2. Do you want to recheck the rows that seem contradictory? 3. Would you like me to solve specific sub-parts or only focus on $$w, x, y, z$$? 4. Should I double-check the entire equation with alternate assumptions? 5. Do you want an explanation of why matrix consistency matters? ### Tip: Always double-check the alignment of equations in matrix problems to ensure no misalignment in values!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Equations
Formulas
Element-wise equality for matrices
Theorems
Equality of matrices requires all corresponding elements to be equal.
Suitable Grade Level
Undergraduate (Linear Algebra Course)