Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.

Question

Emma is a server at an all-you-can eat sushi restaurant. At one table, the customers ordered 3 child buffets and 2 adult buffets, which cost a total of $91. At another table, the customers ordered 3 child buffets and 4 adult buffets, paying a total of $131. How much does the buffet cost for each child and adult?

The cost for a child is $
, and the cost for an adult is $
.

Answer 1

Let \( c \) represent the cost of a child buffet, and \( a \) represent the cost of an adult buffet. We can set up a system of equations based on the information given:

1. From the first table with 3 child buffets and 2 adult buffets totaling $91: \[ 3c + 2a = 91 \]

2. From the second table with 3 child buffets and 4 adult buffets totaling $131: \[ 3c + 4a = 131 \]

Now we have the following system of equations: \[ \begin{aligned}

1. & \quad 3c + 2a = 91 \
2. & \quad 3c + 4a = 131 \end{aligned} \]

Next, we can solve this system. We can eliminate \( c \) by subtracting equation 1 from equation 2:

\[ (3c + 4a) - (3c + 2a) = 131 - 91 \]

This simplifies to: \[ 2a = 40 \]

Now, divide both sides by 2 to solve for \( a \): \[ a = 20 \]

Next, we substitute the value of \( a \) back into equation 1 to find \( c \): \[ 3c + 2(20) = 91 \] \[ 3c + 40 = 91 \]

Subtract 40 from both sides: \[ 3c = 51 \]

Now, divide by 3: \[ c = 17 \]

Thus, the cost for a child buffet is \( $17 \) and the cost for an adult buffet is \( $20 \).

We can fill in the blanks: The cost for a child is $17, and the cost for an adult is $20.