Math Problem Statement
What is the value of a in the matrix equation given in the image?
Solution
The matrix equality provided 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}$$ To solve for $$a$$, let's equate the elements: ### Step 1: Equating the first elements of both matrices: $$\frac{a+b}{9} = \frac{b+a}{9}$$ This equation is always true, so it doesn't provide new information. ### Step 2: Equating the second elements: $$\frac{2-a}{5-c} = \frac{2a+8}{-c+5}$$ We know that $$5 - c = -c + 5$$, so this simplifies to: $$2 - a = 2a + 8$$ ### Step 3: Solve for $$a$$: $$2 - a = 2a + 8$$ $$2 - 8 = 2a + a$$ $$-6 = 3a$$ $$a = -2$$ Thus, the value of $$a$$ is **-2**. Would you like further clarifications? Let me provide related questions and a tip for deeper understanding: --- ### Related Questions: 1. How do we confirm the validity of a matrix equality? 2. What happens if $$c$$ causes a denominator to become zero? 3. How does matrix element-by-element comparison work? 4. Can matrices of different dimensions be equated? 5. How do changes in $$c$$ affect the solution for $$a$$? --- ### Tip: Always check for potential constraints or undefined values (e.g., $$c = 5$$ leading to a denominator of zero) when solving equations involving fractions.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Equality
Linear Equations
Formulas
Matrix element equality: A[i,j] = B[i,j]
Theorems
Equality of matrices requires all corresponding elements to be equal.
Suitable Grade Level
Grades 10-12