Write a system of equations to describe the situation below, solve using any method, and fill in the blanks. $\newline$ Mrs. Moore is researching what it would cost to order flower arrangements for a fancy party. She wants one large centerpiece for the head table, and smaller arrangements for the smaller tables. Bloomington Florist charges $\$10$ for each smaller arrangement, plus $\$45$ for the large centerpiece. Quincy's Flowers, in contrast, charges $\$47$ for the large centerpiece and $\$9$ per arrangement for the rest. If Mrs. Moore orders a certain number of small arrangements, the cost will be the same at either flower shop. How many small arrangements would that be? What would the total cost be? $\newline$ If Mrs. Moore orders _ small arrangements, it will cost $\$$ _ at either shop.

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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

Mrs. Moore is researching what it would cost to order flower arrangements for a fancy party. She wants one large centerpiece for the head table, and smaller arrangements for the smaller tables. Bloomington Florist charges

\$10

for each smaller arrangement, plus

\$45

for the large centerpiece. Quincy's Flowers, in contrast, charges

\$47

for the large centerpiece and

\$9

per arrangement for the rest. If Mrs. Moore orders a certain number of small arrangements, the cost will be the same at either flower shop. How many small arrangements would that be? What would the total cost be?

\newline

If Mrs. Moore orders _____ small arrangements, it will cost

\$

_____ at either shop.

Set up equations: Set up the equations based on the given information. $\newline$ Bloomington Florist: Cost = $\$10$ per small arrangement + $\$45$ for large centerpiece. $\newline$ Quincy's Flowers: Cost = $\$9$ per small arrangement + $\$47$ for large centerpiece. $\newline$ Let $x$ be the number of small arrangements and $C$ be the total cost at either shop. $\newline$ Equation for Bloomington Florist: $10x + 45 = C$ $\newline$ Equation for Quincy's Flowers: $9x + 47 = C$
Equalize equations: Since the costs are the same, we can set the two equations equal to each other to find the number of small arrangements. $10x + 45 = 9x + 47$
Solve for x: Solve for x by subtracting $9x$ from both sides of the equation. $\newline$ $10x - 9x + 45 = 9x - 9x + 47$ $\newline$ $x + 45 = 47$
Substitute $x$ : Subtract $45$ from both sides to solve for $x$ . $\newline$ $x + 45 - 45 = 47 - 45$ $\newline$ $x = 2$
Find total cost: Now that we have the number of small arrangements, we can find the total cost by substituting $x$ back into either original equation. $\newline$ Using Bloomington Florist's equation: $10x + 45 = C$ $\newline$ $10(2) + 45 = C$ $\newline$ $20 + 45 = C$ $\newline$ $C = 65$
Verify solution: Verify the solution by substituting $x$ into Quincy's Flowers' equation. $9x + 47 = C$ $9(2) + 47 = C$ $18 + 47 = C$ $C = 65$