Write a system of equations to describe the situation below, solve using any method, and fill in the blanks. $\newline$ Mrs. Coleman 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. Norwood Florist charges $\$16$ for each smaller arrangement, plus $\$33$ for the large centerpiece. Adam's Flowers, in contrast, charges $\$41$ for the large centerpiece and $\$8$ per arrangement for the rest. If Mrs. Coleman orders a certain number of small arrangements, the cost will be the same at either flower shop. What would the total cost be? How many small arrangements would that be? $\newline$ The cost will be $\$\_\_\_\_$ if Mrs. Coleman orders $\_\_\_$ of the small arrangements.

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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. Coleman 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. Norwood Florist charges

\$16

for each smaller arrangement, plus

\$33

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

\$41

for the large centerpiece and

\$8

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

\newline

The cost will be

\$\_\_\_\_

if Mrs. Coleman orders

\_\_\_

of the small arrangements.

Define Costs and Variables: Let's denote the cost of the large centerpiece from Norwood Florist as $N$ and from Adam's Flowers as $A$ . Let's denote the cost per smaller arrangement from Norwood Florist as $n$ and from Adam's Flowers as $a$ . Let's denote the number of small arrangements as $x$ . We are given that $N = \$33$ , $n = \$16$ , $A = \$41$ , and $a = \$8$ . We need to find the total cost and the number of small arrangements ( $x$ ) for which the cost is the same at both flower shops.
Norwood's Total Cost Equation: Write the equation for the total cost at Norwood Florist, which is the cost of the large centerpiece plus the cost of $x$ small arrangements. $\newline$ Norwood's total cost equation: $33 + 16x$
Adam's Total Cost Equation: Write the equation for the total cost at Adam's Flowers, which is the cost of the large centerpiece plus the cost of $x$ small arrangements. $\newline$ Adam's total cost equation: $41 + 8x$
Set Equations Equal: Since the total cost is the same at both flower shops, we can set the two equations equal to each other to find the value of $x$ . $33 + 16x = 41 + 8x$
Solve for x: Solve for x by subtracting $8x$ from both sides of the equation. $\newline$ $33 + 16x - 8x = 41 + 8x - 8x$ $\newline$ $33 + 8x = 41$
Isolate Variable $x$ : Subtract $33$ from both sides of the equation to isolate the variable $x$ . $33 + 8x - 33 = 41 - 33$ $8x = 8$
Calculate Total Cost: Divide both sides of the equation by $8$ to solve for $x$ . $\newline$ $\frac{8x}{8} = \frac{8}{8}$ $\newline$ $x = 1$
Calculate Total Cost: Divide both sides of the equation by $8$ to solve for $x$ . $\frac{8x}{8} = \frac{8}{8}$ $x = 1$ Now that we have the number of small arrangements ( $x$ ), we can calculate the total cost by substituting $x$ back into either of the total cost equations. We'll use Norwood's equation for this calculation. $\text{Total cost} = 33 + 16(1)$ $\text{Total cost} = 33 + 16$ $\text{Total cost} = 49$