Write a system of equations to describe the situation below, solve using any method, and fill in the blanks. $\newline$ The owner of Arlington Florist is assembling flower arrangements for Valentine's Day. This morning, she assembled $11$ small arrangements and $11$ large arrangements, which took her a total of $88$ minutes. After lunch, she arranged $8$ small arrangements and $14$ large arrangements, which took $100$ minutes. How long does it take to assemble each type? $\newline$ The florist can assemble a small arrangement in _ minutes and a large one in _ minutes.

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Solve a system of equations using any method: word problems

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

\newline

The owner of Arlington Florist is assembling flower arrangements for Valentine's Day. This morning, she assembled

11

small arrangements and

11

large arrangements, which took her a total of

88

minutes. After lunch, she arranged

8

small arrangements and

14

large arrangements, which took

100

minutes. How long does it take to assemble each type?

\newline

The florist can assemble a small arrangement in _____ minutes and a large one in _____ minutes.

Define Equations: Let's denote the time it takes to assemble a small arrangement as $x$ minutes and a large arrangement as $y$ minutes. The first equation is based on the morning session where the florist assembled $11$ small arrangements and $11$ large arrangements in $88$ minutes. $\newline$ First equation: $11x + 11y = 88$
Morning Session: The second equation is based on the afternoon session where the florist assembled $8$ small arrangements and $14$ large arrangements in $100$ minutes. $\newline$ Second equation: $8x + 14y = 100$
Afternoon Session: We have a system of equations: $\newline$ $11x + 11y = 88$ $\newline$ $8x + 14y = 100$ $\newline$ To solve the system, we can simplify the first equation by dividing all terms by $11$ to make the coefficients easier to work with. $\newline$ $x + y = 8$
Simplify Equations: Now we have a simplified system of equations: $\newline$ $x + y = 8$ $\newline$ $8x + 14y = 100$ $\newline$ We can use the substitution or elimination method to solve this system. Let's use the elimination method by multiplying the first equation by $-8$ to eliminate $x$ . $\newline$ $-8(x + y) = -8(8)$ $\newline$ $-8x - 8y = -64$
Elimination Method: Add the new equation to the second equation to eliminate $x$ : $-8x - 8y = -64$ $8x + 14y = 100$ Adding these equations gives us: $6y = 36$
Solve for y: Divide both sides of the equation by $6$ to solve for y: $\newline$ $\frac{6y}{6} = \frac{36}{6}$ $\newline$ $y = 6$ $\newline$ This means it takes $6$ minutes to assemble a large arrangement.
Solve for x: Substitute the value of $y$ back into the simplified first equation to solve for $x$ : $x + 6 = 8$ $x = 8 - 6$ $x = 2$ This means it takes $2$ minutes to assemble a small arrangement.
Final Values: We have found the values for $x$ and $y$ : $\newline$ $x = 2$ (time to assemble a small arrangement) $\newline$ $y = 6$ (time to assemble a large arrangement) $\newline$ These values answer the question prompt.