Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks. $\newline$ Mrs. Kerr 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. Lakeside Florist charges $\$13$ for each smaller arrangement, plus $\$44$ for the large centerpiece. Emily's Flowers, in contrast, charges $\$40$ for the large centerpiece and $\$17$ per arrangement for the rest. If Mrs. Kerr 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. Kerr orders $\_\_\_$ of the small arrangements.

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Solve a system of equations using substitution: word problems

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

\newline

Mrs. Kerr 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. Lakeside Florist charges

\$13

for each smaller arrangement, plus

\$44

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

\$40

for the large centerpiece and

\$17

per arrangement for the rest. If Mrs. Kerr 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. Kerr orders

\_\_\_

of the small arrangements.

Define Variables: Let $x$ represent the number of small arrangements Mrs. Kerr orders. The total cost at Lakeside Florist would be the cost of the large centerpiece plus the cost of $x$ small arrangements. Therefore, the equation for Lakeside Florist is: $\newline$ Cost = $\$44$ (large centerpiece) + $\$13x$ (small arrangements) $\newline$ Cost = $44 + 13x$
Equation for Lakeside Florist: Similarly, the total cost at Emily's Flowers would be the cost of the large centerpiece plus the cost of $x$ small arrangements. The equation for Emily's Flowers is: $\newline$ Cost = $\$40$ (large centerpiece) + $\$17x$ (small arrangements) $\newline$ Cost = $40 + 17x$
Equation for Emily's Flowers: Since the cost will be the same at either flower shop when Mrs. Kerr orders a certain number of small arrangements, we can set the two equations equal to each other to find the value of $x$ : $44 + 13x = 40 + 17x$
Set Equations Equal: Now we solve for $x$ by subtracting $13x$ from both sides of the equation: $\newline$ $44 + 13x - 13x = 40 + 17x - 13x$ $\newline$ $44 = 40 + 4x$
Solve for x: Next, we subtract $40$ from both sides of the equation to isolate the term with $x$ : $\newline$ $44 - 40 = 40 + 4x - 40$ $\newline$ $4 = 4x$
Isolate x Term: Finally, we divide both sides by $4$ to solve for x: $\newline$ $\frac{4}{4} = \frac{4x}{4}$ $\newline$ $1 = x$
Final Solution: Now that we know $x = 1$ , we can find the total cost by substituting $x$ back into either of the original equations. We'll use the Lakeside Florist equation: $\newline$ Cost = $44 + 13x$ $\newline$ Cost = $44 + 13(1)$ $\newline$ Cost = $44 + 13$ $\newline$ Cost = $57$