Question

m.14 solve equations with variables on both sides: word probl...
dan is planning a fancy dinner for his friends birthday. heirloom café charges $21 per person for the meal, plus an additional $8 per person for dessert. hot stone bakery charges $26 per person for the meal, plus $33 for a birthday cake the friends can share.
which equation can you use to find p, the number of people who would need to come to the dinner for the two restaurants to cost the same?
$26p + 8 = 21p + 33$
$21p + 8p = 26p + 33$
how many people would need to come to the dinner for the two restaurants to cost the same?
\boxed{} people

Explanation:

First, determine the correct equation:

• For Heirloom Café: The cost per person for the meal is $21, and an additional $8 per person for dessert. So the total cost for \( p \) people is \( (21 + 8)p=21p + 8p \).
• For Hot Stone Bakery: The cost per person for the meal is $26, and a fixed $33 for the birthday cake. So the total cost for \( p \) people is \( 26p+33 \).
• To find when the costs are the same, we set the two expressions equal: \( 21p + 8p=26p + 33 \).

Then, solve the equation \( 21p + 8p=26p + 33 \):

Step 1: Combine like terms on the left side

\( 21p+8p=(21 + 8)p = 29p \), so the equation becomes \( 29p=26p + 33 \).

Step 2: Subtract \( 26p \) from both sides

\( 29p-26p=26p + 33-26p \), which simplifies to \( 3p=33 \).

Step 3: Divide both sides by 3

\( \frac{3p}{3}=\frac{33}{3} \), so \( p = 11 \).

Answer:

The correct equation is \( 21p + 8p = 26p + 33 \), and the number of people is \( \boldsymbol{11} \).