Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks. $\newline$ An online boutique is having a special on personalized baby items. On Monday, they sold $12$ personalized baby blankets and $11$ personalized hooded towels, for a total of $\$569$ in receipts. The following day, they received orders for $1$ personalized baby blanket and $11$ personalized hooded towels, which brought in a total of $\$239$ . How much does each item sell for? $\newline$ Blankets sell for $\$$ _ apiece, and hooded towels sell for $\$$ _ apiece.

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Algebra 2

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 online boutique is having a special on personalized baby items. On Monday, they sold

12

personalized baby blankets and

11

personalized hooded towels, for a total of

\$569

in receipts. The following day, they received orders for

1

personalized baby blanket and

11

personalized hooded towels, which brought in a total of

\$239

. How much does each item sell for?

\newline

Blankets sell for

\$

_____ apiece, and hooded towels sell for

\$

_____ apiece.

Define Prices: Let's denote the price of each personalized baby blanket as $b$ and the price of each personalized hooded towel as $t$ . From the first day's sales, we have the equation $12b + 11t = 569$ . This represents the sale of $12$ blankets and $11$ towels for a total of $\$569$ .
First Day Sales: From the second day's sales, we have the equation $1b + 11t = 239$ . This represents the sale of $1$ blanket and $11$ towels for a total of $\$239$ .
Second Day Sales: We now have a system of two equations. To solve using elimination, we need to eliminate one of the variables, $b$ or $t$ . We choose to eliminate $b$ because it has a coefficient of $1$ in the second equation, which makes it easier to manipulate.
Eliminate Variable: To eliminate $b$ , we multiply the second equation by $-12$ , the negative of the coefficient of $b$ in the first equation. This gives us the new equation $-12b - 132t = -2868$ .
Solve for t: We now add the first equation to the new second equation to eliminate $b$ . This gives us $-121t = -2299$ . Solving for $t$ , we get $t = \frac{2299}{121}$ .
Calculate t: After calculating $t = \frac{2299}{121}$ , we find that $t = 19$ . This means that each personalized hooded towel sells for $\$19$ .
Substitute and Solve for b: We substitute $t = 19$ into the second original equation $1b + 11t = 239$ and solve for b. This gives us $1b + 11(19) = 239$ , which simplifies to $1b + 209 = 239$ .
Substitute and Solve for b: We substitute $t = 19$ into the second original equation $1b + 11t = 239$ and solve for b. This gives us $1b + 11(19) = 239$ , which simplifies to $1b + 209 = 239$ . Subtracting $209$ from both sides of the equation $1b + 209 = 239$ , we get $1b = 30$ . Therefore, $b = 30$ . This means that each personalized baby blanket sells for $\$30$ .