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

Question

Mrs. Sandoval 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. Livingston Florist charges $18 for each smaller arrangement, plus $37 for the large centerpiece. Jared's Flowers, in contrast, charges $31 for the large centerpiece and $19 per arrangement for the rest. If Mrs. Sandoval 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?

If Mrs. Sandoval orders
small arrangements, it will cost $
at either shop.

Answer 1

Let's represent the number of small arrangements Mrs. Sandoval orders as \( x \).

1. Cost at Livingston Florist:

   • Cost for one large centerpiece: $37
   • Cost for \( x \) smaller arrangements: $18x
   • Total cost at Livingston Florist: \[ C_L = 37 + 18x \]

2. Cost at Jared's Flowers:

   • Cost for one large centerpiece: $31
   • Cost for \( x \) smaller arrangements: $19x
   • Total cost at Jared's Flowers: \[ C_J = 31 + 19x \]

Since the cost will be the same at either flower shop, we can set the two equations equal to each other: \[ 37 + 18x = 31 + 19x \]

Now, let's solve for \( x \).

3. Rearranging the equation:

   • Subtract \( 18x \) from both sides: \[ 37 = 31 + x \]

4. Subtracting 31 from both sides: \[ 37 - 31 = x \] \[ 6 = x \]

So, Mrs. Sandoval needs to order 6 small arrangements.

5. Calculating the total cost:
   • Substitute \( x = 6 \) back into either cost equation. We'll use the Livingston Florist's equation: \[ C_L = 37 + 18(6) = 37 + 108 = 145 \]

Thus, if Mrs. Sandoval orders 6 small arrangements, it will cost $145 at either shop.

To summarize, the answer is:

If Mrs. Sandoval orders 6 small arrangements, it will cost $145 at either shop.