Algebra II Chapter 8 math worksheet with multiple-choice questions on systems of equations and matrices.

Problem Analysis:

The image contains a series of multiple-choice questions related to solving systems of linear equations. Let's solve each question step by step.

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Question 1: Which is a solution of this system?

$$
\begin{cases}
2x + y = 7 \\
3x - y = 8
\end{cases}
$$

#### Step 1: Solve the system of equations.
We are given:
1. \( 2x + y = 7 \)
2. \( 3x - y = 8 \)

To eliminate \( y \), we can add the two equations:
$$
(2x + y) + (3x - y) = 7 + 8
$$
$$
5x = 15
$$
$$
x = 3
$$

Now substitute \( x = 3 \) into the first equation \( 2x + y = 7 \):
$$
2(3) + y = 7
$$
$$
6 + y = 7
$$
$$
y = 1
$$

Thus, the solution to the system is \( (x, y) = (3, 1) \).

#### Step 2: Match the solution with the options.
The options are:
A. \( (1, 5) \)
B. \( (3, -2) \)
C. \( (3, 1) \)
D. No solution

The correct answer is:
$$
\boxed{C}
$$

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Question 2: The product of \( 3 \sqrt{2} \) and \( 2 \sqrt{5} \) is:

#### Step 1: Simplify the product.
The product is:
$$
3 \sqrt{2} \cdot 2 \sqrt{5}
$$
Using the property of multiplication of square roots, \( \sqrt{a} \cdot \sqrt{b} = \sqrt{ab} \), we get:
$$
(3 \cdot 2) \cdot (\sqrt{2} \cdot \sqrt{5}) = 6 \sqrt{10}
$$

#### Step 2: Match the result with the options.
The options are:
A. \( 6 \sqrt{10} \)
B. \( 6 \sqrt{7} \)
C. \( 5 \sqrt{10} \)
D. \( 5 \sqrt{7} \)

The correct answer is:
$$
\boxed{A}
$$

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Question 3: What does the determinant mean for a \( 2 \times 2 \) matrix?

#### Step 1: Recall the formula for the determinant of a \( 2 \times 2 \) matrix.
For a \( 2 \times 2 \) matrix:
$$
A = \begin{bmatrix}
a & b \\
c & d
\end{bmatrix}
$$
the determinant is given by:
$$
\det(A) = ad - bc
$$

#### Step 2: Match the formula with the options.
The options are:
A. \( \frac{ad}{bc} \)
B. \( \frac{ac}{bd} \)
C. \( ad - bc \)
D. \( ac - bd \)

The correct answer is:
$$
\boxed{C}
$$

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Question 4: Which of the following could be used to find \( z \) in the system?

$$
\begin{cases}
2x + y + z = 13 \\
3x - y + z = 12
\end{cases}
$$

#### Step 1: Analyze the system.
We are given:
1. \( 2x + y + z = 13 \)
2. \( 3x - y + z = 12 \)

To isolate \( z \), we can eliminate \( y \) by adding the two equations:
$$
(2x + y + z) + (3x - y + z) = 13 + 12
$$
$$
5x + 2z = 25
$$
$$
2z = 25 - 5x
$$
$$
z = \frac{25 - 5x}{2}
$$

#### Step 2: Match the expression for \( z \) with the options.
The options are:
A. \( z = \frac{25 - 5x}{2} \)
B. \( z = \frac{25 - 5y}{2} \)
C. \( z = \frac{25 - 3x}{2} \)
D. \( z = \frac{25 - 3y}{2} \)

The correct answer is:
$$
\boxed{A}
$$

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Final Answers:

1. \( \boxed{C} \)
2. \( \boxed{A} \)
3. \( \boxed{C} \)
4. \( \boxed{A} \)