Question

exercise

1. alice is selling lemonade and fresh-squeezed orange juice at a booth. she sells cups of lemonade for \\$1 each and cups of orange juice for \\$3 each. write an expression that represents the amount of money that alice earns selling drinks.

1. evaluate \\(2x - 4\\) for \\(x = 9\\).

1. evaluate \\(-b + 8\\) for \\(b = 3\\).

1. find the value of the expression \\(\frac{1}{2}z + 3\\) if \\(x = 12\\).

1. for the equation \\(y = 3x + 4\\), find the value of \\(y\\) if \\(x = 2\\).

1. the expression \\(\\$25 + \\$10h\\) gives the amount of money that stephanie earns per day, where \\(h\\) is the number of hours she works. how much does she earn if she works 4 hours?

1. simplify each of the following expressions:

a. \\(3(x + 2) + 4(x - 2)\\)
b. \\(2(x + y) - 3(x + y)\\)
c. \\(-(y + 8) + 4(6 - y)\\)
d. \\(2(4x + 3) - 3(x + 1)\\)

Explanation:

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<reasoning>

Define variables and write linear model

Let \(L\) be the number of cups of lemonade sold and \(O\) be the number of cups of orange juice sold.
\[
\text{Earnings} = 1 \cdot L + 3 \cdot O = L + 3O
\]

Evaluate expressions for given values

For Question 2:
\[
2x - 4 \text{ at } x = 9 \implies 2(9) - 4 = 18 - 4 = 14
\]
For Question 3:
\[
-b + 8 \text{ at } b = 3 \implies -3 + 8 = 5
\]
For Question 4 (assuming the variable in the expression is \(x\) instead of \(z\) due to the typo "if \(x = 12\)"):
\[
\frac{1}{2}x + 3 \text{ at } x = 12 \implies \frac{1}{2}(12) + 3 = 6 + 3 = 9
\]
For Question 5:
\[
y = 3x + 4 \text{ at } x = 2 \implies y = 3(2) + 4 = 6 + 4 = 10
\]
For Question 6:
\[
25 + 10h \text{ at } h = 4 \implies 25 + 10(4) = 25 + 40 = 65
\]

Simplify algebraic expressions

For Question 7:
a. \(3(x + 2) + 4(x - 2) = 3x + 6 + 4x - 8 = 7x - 2\)
b. \(2(x + y) - 3(x + y) = 2x + 2y - 3x - 3y = -x - y\)
c. \(-(y + 8) + 4(6 - y) = -y - 8 + 24 - 4y = -5y + 16\)
d. \(2(4x + 3) - 3(x + 1) = 8x + 6 - 3x - 3 = 5x + 3\)
</reasoning>

<answer>

No.	Answer
2	\(14\)
3	\(5\)
4	\(9\)
5	\(10\)
6	\(\$65\)
7a	\(7x - 2\)
7b	\(-x - y\)
7c	\(-5y + 16\)
7d	\(5x + 3\)

</answer>

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Answer:

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<reasoning>

Define variables and write linear model

Let \(L\) be the number of cups of lemonade sold and \(O\) be the number of cups of orange juice sold.
\[
\text{Earnings} = 1 \cdot L + 3 \cdot O = L + 3O
\]

Evaluate expressions for given values

For Question 2:
\[
2x - 4 \text{ at } x = 9 \implies 2(9) - 4 = 18 - 4 = 14
\]
For Question 3:
\[
-b + 8 \text{ at } b = 3 \implies -3 + 8 = 5
\]
For Question 4 (assuming the variable in the expression is \(x\) instead of \(z\) due to the typo "if \(x = 12\)"):
\[
\frac{1}{2}x + 3 \text{ at } x = 12 \implies \frac{1}{2}(12) + 3 = 6 + 3 = 9
\]
For Question 5:
\[
y = 3x + 4 \text{ at } x = 2 \implies y = 3(2) + 4 = 6 + 4 = 10
\]
For Question 6:
\[
25 + 10h \text{ at } h = 4 \implies 25 + 10(4) = 25 + 40 = 65
\]

Simplify algebraic expressions

For Question 7:
a. \(3(x + 2) + 4(x - 2) = 3x + 6 + 4x - 8 = 7x - 2\)
b. \(2(x + y) - 3(x + y) = 2x + 2y - 3x - 3y = -x - y\)
c. \(-(y + 8) + 4(6 - y) = -y - 8 + 24 - 4y = -5y + 16\)
d. \(2(4x + 3) - 3(x + 1) = 8x + 6 - 3x - 3 = 5x + 3\)
</reasoning>

<answer>

No.	Answer
2	\(14\)
3	\(5\)
4	\(9\)
5	\(10\)
6	\(\$65\)
7a	\(7x - 2\)
7b	\(-x - y\)
7c	\(-5y + 16\)
7d	\(5x + 3\)

</answer>

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