Math Problem Statement
Solve for x and y in the given matrix equation:
\[
\begin{pmatrix}
-8 & 7 & 1 \\
x & x+2 & -2
\end{pmatrix}
-
\begin{pmatrix}
2y + 1 & 6 & 6 \\
1 & 5 & y
\end{pmatrix}
=
\begin{pmatrix}
-7 & -5 & 6 \\
-7 & 2 & 6
\end{pmatrix}
\]
Solution
The given problem is a matrix equation involving two matrices that are set equal to each other. The task is to find the values of and that satisfy the matrix equation. The matrices are:
-8 & 7 & 1 \\ x & x+2 & -2 \end{pmatrix} - \begin{pmatrix} 2y + 1 & 6 & 6 \\ 1 & 5 & y \end{pmatrix} = \begin{pmatrix} -7 & -5 & 6 \\ -7 & 2 & 6 \end{pmatrix}$$ ### Step 1: Write out the matrix subtraction Let's subtract the second matrix from the first matrix element-wise: $$\begin{pmatrix} -8 & 7 & 1 \\ x & x+2 & -2 \end{pmatrix} - \begin{pmatrix} 2y + 1 & 6 & 6 \\ 1 & 5 & y \end{pmatrix} = \begin{pmatrix} (-8 - (2y + 1)) & (7 - 6) & (1 - 6) \\ (x - 1) & ((x+2) - 5) & (-2 - y) \end{pmatrix}$$ ### Step 2: Simplify each element of the resulting matrix Simplifying each term: $$\begin{pmatrix} -8 - (2y + 1) & 7 - 6 & 1 - 6 \\ x - 1 & (x + 2 - 5) & -2 - y \end{pmatrix} = \begin{pmatrix} -9 - 2y & 1 & -5 \\ x - 1 & x - 3 & -2 - y \end{pmatrix}$$ ### Step 3: Set up the element-wise equations We now compare this resulting matrix to the matrix on the right-hand side: $$\begin{pmatrix} -9 - 2y & 1 & -5 \\ x - 1 & x - 3 & -2 - y \end{pmatrix} = \begin{pmatrix} -7 & -5 & 6 \\ -7 & 2 & 6 \end{pmatrix}$$ This gives us the following system of equations by equating corresponding elements: 1. $$-9 - 2y = -7$$ 2. $$1 = -5$$ (which results in a contradiction) 3. $$-5 = 6$$ (which also results in a contradiction) 4. $$x - 1 = -7$$ 5. $$x - 3 = 2$$ 6. $$-2 - y = 6$$ ### Step 4: Solve for $$y$$ and $$x$$ Let's solve the valid equations: - From equation 1: $$-9 - 2y = -7$$ $$-2y = -7 + 9 = 2 \quad \Rightarrow \quad y = -1$$ - From equation 4: $$x - 1 = -7$$ $$x = -7 + 1 = -6$$ - From equation 5: $$x - 3 = 2$$ $$x = 2 + 3 = 5$$ There seems to be an inconsistency between these equations for $$x$$. This suggests that either there is no solution or the system is overdetermined. However, for the purpose of this exercise, it seems we only needed to solve for $$x$$ and $$y$$, assuming one of the two solutions for $$x$$ is valid depending on the context. Would you like more details or further clarification? Here are some additional questions to explore: 1. How do you handle inconsistent systems of equations? 2. What conditions make two matrices equal? 3. How can matrix subtraction be applied in solving systems of equations? 4. What other methods can solve for unknowns in matrix equations? 5. What does it mean when a system of equations has no solution? **Tip:** Always check each step when solving systems of equations to ensure that contradictions don't go unnoticed.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Subtraction
Systems of Equations
Linear Algebra
Formulas
Matrix subtraction: A - B = C where element-wise subtraction occurs.
Solving systems of equations using matrix elements
Theorems
Equality of Matrices Theorem
Suitable Grade Level
Grades 11-12