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 Lakewood, she ordered $4$ small ice sculptures and $5$ large ice sculptures, which cost $\$871$ . Then, for a release party in Danville, she ordered $5$ small ice sculptures and $2$ large ice sculptures, which cost a total of $\$549$ . How much does each ice sculpture cost? $\newline$ Each small ice sculpture costs $\$$ , and each large one costs $\$$ _.

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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

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

4

small ice sculptures and

5

large ice sculptures, which cost

\$871

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

5

small ice sculptures and

2

large ice sculptures, which cost a total of

\$549

. How much does each ice sculpture cost?

\newline

Each small ice sculpture costs

\$

____, and each large one costs

\$

_____.

Equations Setup: Let's denote the cost of a small ice sculpture as $S$ and the cost of a large ice sculpture as $L$ . We can then write two equations based on the information given: $\newline$ For the executive dinner in Lakewood: $4S + 5L = \$871$ $\newline$ For the release party in Danville: $5S + 2L = \$549$
Elimination Method: To solve this system using elimination, we want to eliminate one of the variables. We can do this by multiplying the second equation by $2$ , so that the coefficient of $L$ in both equations is the same: $\newline$ $2\times(5S + 2L) = 2\times(\$549)$ $\newline$ This gives us: $10S + 4L = (\$1098)$
New System of Equations: Now we have the system of equations: $\newline$ $4S + 5L = \$(871)$ $\newline$ $10S + 4L = \$(1098)$ $\newline$ We can eliminate $L$ by multiplying the first equation by $4$ and the second equation by $5$ and then subtracting the second equation from the first: $\newline$ $4*(4S + 5L) = 4*\$(871)$ $\newline$ $5*(10S + 4L) = 5*\$(1098)$ $\newline$ This gives us: $16S + 20L = \$(3484)$ $\newline$ and: $50S + 20L = \$(5490)$
Solving for S: Subtract the second equation from the first: $\newline$ $(16S + 20L) - (50S + 20L) = (\$)3484 - (\$)5490$ $\newline$ $-34S = -(\$)2006$ $\newline$ Now, divide both sides by $-34$ to solve for S: $\newline$ $S = (\$)2006 / 34$ $\newline$ $S = (\$)59$
Substitute S to Find L: Now that we have the value for S, we can substitute it back into one of the original equations to solve for L. Let's use the first equation: $\newline$ $4S + 5L = \$871$ $\newline$ $4*\$59 + 5L = \$871$ $\newline$ $\$236 + 5L = \$871$ $\newline$ $5L = \$871 - \$236$ $\newline$ $5L = \$635$ $\newline$ Now, divide both sides by $5$ to solve for L: $\newline$ $L = \$635 / 5$ $\newline$ $L = \$127$