Question

jenna has one more logic puzzle to solve!

3 \text{ books} + 2 \text{ music notes} = 33
1 \text{ book} + 3 \text{ music notes} = 25

write a system of equations to represent this puzzle, using \\(b\\) to represent the value of each book and \\(m\\) to represent the value of each music note. then solve the system for \\(b\\) and \\(m\\).

type the correct answer in each box. use numerals instead of words.

the value of each book is .
the value of each music note is .

Explanation:

⚡ Using what you learned: introduction to systems of equations

Step 1: Write the system of equations

Translate the visual representations into algebraic equations using \( b \) for books and \( m \) for music notes:

• First line: 3 books and 2 music notes equal 33.

$$ 3b + 2m = 33 $$

• Second line: 1 book and 3 music notes equal 25.

$$ b + 3m = 25 $$

Step 2: Solve for \( b \) in terms of \( m \)

Using the second equation:

$$ b = 25 - 3m $$

Step 3: Substitute and solve for \( m \)

Substitute \( b \) into the first equation:

$$ 3(25 - 3m) + 2m = 33 $$

$$ 75 - 9m + 2m = 33 $$

$$ 75 - 7m = 33 $$

$$ -7m = -42 $$

$$ m = 6 $$

Step 4: Solve for \( b \)

Substitute \( m = 6 \) back into the expression for \( b \):

$$ b = 25 - 3(6) $$

$$ b = 25 - 18 $$

$$ b = 7 $$

Answer:

The value of each book is 7.

The value of each music note is 6.