Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks. $\newline$ Carson is a server at an all-you-can eat sushi restaurant. At one table, the customers ordered $1$ child buffet and $4$ adult buffets, which cost a total of $\$110$ . At another table, the customers ordered $3$ child buffets and $3$ adult buffets, paying a total of $\$114$ . How much does the buffet cost for each child and adult? $\newline$ The cost for a child is $\$\_$ , and the cost for an adult is $\$\_$ .

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Algebra 1

Solve a system of equations using elimination: word problems

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

\newline

Carson is a server at an all-you-can eat sushi restaurant. At one table, the customers ordered

1

child buffet and

4

adult buffets, which cost a total of

\$110

. At another table, the customers ordered

3

child buffets and

3

adult buffets, paying a total of

\$114

. How much does the buffet cost for each child and adult?

\newline

The cost for a child is

\$\_

, and the cost for an adult is

\$\_

Equations Setup: Let's denote the cost of the child buffet as $x$ and the cost of the adult buffet as $y$ . We can write two equations based on the information given. $\newline$ First table: $1$ child buffet + $4$ adult buffets = $\$110$ $\newline$ Second table: $3$ child buffets + $3$ adult buffets = $\$114$ $\newline$ Translate this information into equations: $\newline$ $1x + 4y = 110$ $\newline$ $3x + 3y = 114$
Elimination Method: To use elimination, we need to make the coefficients of one of the variables the same in both equations. We can multiply the first equation by $3$ to match the coefficient of $x$ in the second equation. $\newline$ Multiplying the first equation by $3$ gives us: $\newline$ $3x + 12y = 330$ $\newline$ We now have the system: $\newline$ $3x + 12y = 330$ $\newline$ $3x + 3y = 114$
Subtract Equations: Subtract the second equation from the first equation to eliminate $x$ . $\newline$ $(3x + 12y) - (3x + 3y) = 330 - 114$ $\newline$ This simplifies to: $\newline$ $3x + 12y - 3x - 3y = 330 - 114$ $\newline$ $9y = 216$
Solve for y: Divide both sides of the equation by $9$ to solve for y. $\newline$ $\frac{9y}{9} = \frac{216}{9}$ $\newline$ $y = 24$
Substitute and Solve: Now that we have the value of $y$ , we can substitute it back into one of the original equations to solve for $x$ . Let's use the first equation: $\newline$ $1x + 4(24) = 110$
Final Cost Calculation: Simplify the equation and solve for $x$ . $x + 96 = 110$ $x = 110 - 96$ $x = 14$
Final Cost Calculation: Simplify the equation and solve for $x$ . $x + 96 = 110$ $x = 110 - 96$ $x = 14$ We have found the values of $x$ and $y$ , which represent the cost of the child buffet and the adult buffet, respectively. $x = 14$ $y = 24$ The cost for a child buffet is $\$14$ , and the cost for an adult buffet is $\$24$ .