Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks. $\newline$ A local service organization is wrapping gifts at the mall to raise money for charity. Yesterday, they wrapped $22$ small gifts and $21$ large gifts, earning a total of $\$191$ . Today, they wrapped $18$ small gifts and $10$ large gifts, and earned $\$106$ . How much did they charge to wrap the gifts? $\newline$ The organization charges _ to wrap a small gift and _ to wrap a large one.

Full solution

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

\newline

A local service organization is wrapping gifts at the mall to raise money for charity. Yesterday, they wrapped

22

small gifts and

21

large gifts, earning a total of

\$191

. Today, they wrapped

18

small gifts and

10

large gifts, and earned

\$106

. How much did they charge to wrap the gifts?

\newline

The organization charges _____ to wrap a small gift and _____ to wrap a large one.

Equations setup: Let's denote the charge for wrapping a small gift as $s$ and for a large gift as $l$ . We can write two equations based on the given information: $\newline$ $1$ . For yesterday's earnings: $22s + 21l = 191$ $\newline$ $2$ . For today's earnings: $18s + 10l = 106$ $\newline$ These two equations form our system of equations.
Elimination method: To solve this system using elimination, we need to eliminate one of the variables. We can do this by multiplying the second equation by a number that will make the coefficient of one of the variables the same in both equations. Let's multiply the second equation by $2$ to match the coefficient of $l$ in the first equation: $\newline$ $2 \times (18s + 10l) = 2 \times 106$ $\newline$ This gives us: $\newline$ $36s + 20l = 212$
Coefficient matching: Now we have the system of equations: $\newline$ $1$ . $22s + 21l = 191$ $\newline$ $2$ . $36s + 20l = 212$ $\newline$ We can subtract the first equation from the second to eliminate $l$ : $\newline$ $(36s + 20l) - (22s + 21l) = 212 - 191$ $\newline$ This simplifies to: $\newline$ $14s - l = 21$
Subtraction correction: We made a mistake in the previous step. We should have subtracted the coefficients correctly. Let's correct that: $\newline$ $(36s + 20l) - (22s + 21l) = 212 - 191$ $\newline$ This simplifies to: $\newline$ $14s - l = 21$ $\newline$ This is incorrect. We need to subtract the $l$ terms correctly.