Write a system of equations to describe the situation below, solve using any method, and fill in the blanks. $\newline$ Mrs. Lee 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. Yardley Florist charges $\$7$ for each smaller arrangement, plus $\$46$ for the large centerpiece. Carrie's Flowers, in contrast, charges $\$42$ for the large centerpiece and $\$9$ per arrangement for the rest. If Mrs. Lee 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. Lee 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. Lee 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. Yardley Florist charges

\$7

for each smaller arrangement, plus

\$46

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

\$42

for the large centerpiece and

\$9

per arrangement for the rest. If Mrs. Lee 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. Lee orders _____ small arrangements, it will cost

\$

_____ at either shop.

Set up equations: Set up the equations based on the information given. $\newline$ Yardley Florist charges $\$7$ for each smaller arrangement plus $\$46$ for the large centerpiece. Let $x$ be the number of small arrangements. The total cost at Yardley Florist is then $7x + 46$ . $\newline$ Carrie's Flowers charges $\$9$ for each smaller arrangement plus $\$42$ for the large centerpiece. The total cost at Carrie's Flowers is then $9x + 42$ . $\newline$ Since the costs are the same, we can set the two expressions equal to each other to form the equation: $\newline$ $7x + 46 = 9x + 42$
Solve equation for $x$ : Solve the equation for $x$ . $\newline$ Subtract $7x$ from both sides to get: $\newline$ $46 = 2x + 42$ $\newline$ Subtract $42$ from both sides to get: $\newline$ $4 = 2x$ $\newline$ Divide both sides by $2$ to get: $\newline$ $x = 2$
Calculate total cost: Calculate the total cost for either shop. $\newline$ Now that we know $x = 2$ , we can substitute this value into either shop's cost equation to find the total cost. $\newline$ Using Yardley Florist's cost equation: $\newline$ Total cost = $7(2) + 46 = 14 + 46 = \$60$
Verify total cost: Verify the total cost with Carrie's Flowers to ensure it is the same. $\newline$ Using Carrie's Flowers' cost equation: $\newline$ Total cost = $$9$($2$) + $42$ = $18$ + $42$ = (\$)$60$