Here is a detailed breakdown and solution for the provided questions from your College Algebra test.

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Question 1: Matrices

Given matrices:

• $A = \begin{pmatrix} 4 & 5 \\ 1 & 2 \end{pmatrix}$,
• $B = \begin{pmatrix} 1 & 4 & 0 \\ 5 & 3 & 1 \end{pmatrix}$,
• $C = \begin{pmatrix} 6 & 5 \\ 1 & -3 \end{pmatrix}$.

(a) $3C$: $3C = 3 \cdot \begin{pmatrix} 6 & 5 \\ 1 & -3 \end{pmatrix} = \begin{pmatrix} 18 & 15 \\ 3 & -9 \end{pmatrix}.$

(b) $A + C$: $A + C = \begin{pmatrix} 4 & 5 \\ 1 & 2 \end{pmatrix} + \begin{pmatrix} 6 & 5 \\ 1 & -3 \end{pmatrix} = \begin{pmatrix} 10 & 10 \\ 2 & -1 \end{pmatrix}.$

(c) $AB$: $AB = \begin{pmatrix} 4 & 5 \\ 1 & 2 \end{pmatrix} \cdot \begin{pmatrix} 1 & 4 & 0 \\ 5 & 3 & 1 \end{pmatrix}.$ Step-by-step: $AB = \begin{pmatrix} (4)(1) + (5)(5) & (4)(4) + (5)(3) & (4)(0) + (5)(1) \\ (1)(1) + (2)(5) & (1)(4) + (2)(3) & (1)(0) + (2)(1) \end{pmatrix} = \begin{pmatrix} 29 & 31 & 5 \\ 11 & 10 & 2 \end{pmatrix}.$

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Question 2: Determine if $(2, -5)$ is a solution.

Given:

1. $3x - y = 11$,
2. $x + y = 3$.

Substitute $x = 2, y = -5$ into both equations:

1. $3(2) - (-5) = 6 + 5 = 11$ (True).
2. $2 + (-5) = -3$ (False).

Conclusion: $(2, -5)$ is not a solution.

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Question 3: Solve the system using substitution or addition.

$x + 3y = -1 \quad \text{(1)},$ $2x - y = 5 \quad \text{(2)}.$

Step 1: Solve (1) for $x$: $x = -1 - 3y.$

Step 2: Substitute into (2): $2(-1 - 3y) - y = 5.$ Simplify: $-2 - 6y - y = 5 \implies -7y = 7 \implies y = -1.$

Step 3: Substitute $y = -1$ into $x = -1 - 3y$: $x = -1 - 3(-1) = 2.$

Solution: $(x, y) = (2, -1)$.

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Would you like me to continue solving the remaining questions in detail? Let me know which ones to prioritize or clarify.

Here are five possible follow-up questions for you:

1. What is the determinant of matrix $A$?
2. Could the matrix product $BA$ be computed? Why or why not?
3. How do you graph $f(x) = 3^x$?
4. What transformations occur in $g(x) = -f(x)$?
5. How can logarithmic and exponential equations be related?

Tip: When working with matrices, always verify dimensions to ensure operations like multiplication are valid!