Math Problem Statement
Solve the matrix equation: \[
\begin{bmatrix}
\frac{a+b}{9} & \frac{2-a}{5-c}\end{bmatrix} = \begin{bmatrix}
\frac{b+a}{9} & \frac{2a+8}{-c+5}\end{bmatrix}\]
Solution
The given matrix equation is:
\frac{a+b}{9} & \frac{2-a}{5-c} \end{bmatrix} = \begin{bmatrix} \frac{b+a}{9} & \frac{2a+8}{-c+5} \end{bmatrix}$$ ### Solving for $$a$$: The matrices are equal, so the corresponding elements must be equal. This gives two equations: 1. $$\frac{a+b}{9} = \frac{b+a}{9}$$ 2. $$\frac{2-a}{5-c} = \frac{2a+8}{-c+5}$$ #### Step 1: Solve the first equation $$\frac{a+b}{9} = \frac{b+a}{9}$$ This equation is always true because the left-hand side and the right-hand side are identical. Thus, this equation gives no additional information. #### Step 2: Solve the second equation $$\frac{2-a}{5-c} = \frac{2a+8}{-c+5}$$ Cross-multiply: $$(2-a)(-c+5) = (2a+8)(5-c)$$ Expand both sides: $$-2c + 10 + ac - 5a = 10a + 40 - 2ac - 8c$$ Simplify: $$-2c + 10 + ac - 5a = 10a + 40 - 2ac - 8c$$ Combine like terms: $$ac + 2ac - 5a - 10a = -8c + 2c + 40 - 10$$ $$3ac - 15a = -6c + 30$$ Factor: $$3a(c - 5) = -6(c - 5)$$ Divide through by $$(c-5)$$, assuming $$c \neq 5$$: $$3a = -6$$ Solve for $$a$$: $$a = -2$$ ### Final Answer: $$a = -2$$ Let me know if you need further clarification or additional steps! Here are some related questions to explore: 1. What is the value of $$b$$ in this equation? 2. How does the solution change if $$c = 5$$? 3. How can we verify the solution by substituting back into the original equation? 4. Are there any constraints on $$c$$ or $$b$$? 5. What methods can be used to solve similar matrix equations systematically? **Tip:** Always check the solution by substituting it back into the original equation to confirm its correctness.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Equations
Linear Algebra
Systems of Equations
Formulas
Matrix equality: corresponding elements of equal matrices are equal
Theorems
Equality of matrices
Cross-multiplication in rational expressions
Suitable Grade Level
Grades 9-12