Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks. $\newline$ An event planner routinely orders ice sculptures for the corporate events she plans. For an executive dinner in Richmond, she ordered $2$ small ice sculptures and $3$ large ice sculptures, which cost $\$546$ . Then, for a release party in Bluepoint, she ordered $5$ small ice sculptures and $3$ large ice sculptures, which cost a total of $\$717$ . How much does each ice sculpture cost? $\newline$ Each small ice sculpture costs $\$\_\_\_\_$ , and each large one costs $\$\_\_\_\_\_$ .

View step-by-step help

Home

Math Problems

Precalculus

Solve a system of equations using elimination: word problems

Full solution

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

\newline

An event planner routinely orders ice sculptures for the corporate events she plans. For an executive dinner in Richmond, she ordered

2

small ice sculptures and

3

large ice sculptures, which cost

\$546

. Then, for a release party in Bluepoint, she ordered

5

small ice sculptures and

3

large ice sculptures, which cost a total of

\$717

. How much does each ice sculpture cost?

\newline

Each small ice sculpture costs

\$\_\_\_\_

, and each large one costs

\$\_\_\_\_\_

Define Equations: Let's denote the cost of a small ice sculpture as $S$ and the cost of a large ice sculpture as $L$ . We can write two equations based on the information given: $\newline$ For the executive dinner in Richmond: $2S + 3L = \$546$ $\newline$ For the release party in Bluepoint: $5S + 3L = \$717$
Eliminate Variable: To solve this system using elimination, we want to eliminate one of the variables. We can do this by subtracting the first equation from the second equation to eliminate $L$ : $\newline$ $(5S + 3L) - (2S + 3L) = (\$717) - (\$546)$
Solve for S: Perform the subtraction to eliminate L and solve for S: $\newline$ $5S - 2S + 3L - 3L = (\$)717 - (\$)546$ $\newline$ $3S = (\$)171$
Substitute and Solve: Now, divide both sides by $3$ to find the value of S: $\newline$ $3S \div 3 = (\$171) \div 3$ $\newline$ $S = (\$57)$
Calculate $2S$ : With the value of $S$ found, we can substitute it back into one of the original equations to find $L$ . Let's use the first equation: $\newline$ $2S + 3L = \$546$ $\newline$ $2(\$57) + 3L = \$546$
Solve for L: Calculate the value of $2$ times S: $\newline$ $2 \times \$57 = \$114$ $\newline$ So, the equation becomes: $\newline$ $\$114 + 3L = \$546$
Find Value of L: Subtract $\$114$ from both sides to solve for L: $\newline$ $3L = \$546 - \$114$ $\newline$ $3L = \$432$
Find Value of L: Subtract $\$114$ from both sides to solve for L: $\newline$ $3L = \$546 - \$114$ $\newline$ $3L = \$432$ Now, divide both sides by $3$ to find the value of L: $\newline$ $3L \div 3 = \$432 \div 3$ $\newline$ $L = \$144$