Mrs. Williamson 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. Silvergrove Florist charges $14 for each smaller arrangement, plus $29 for the large centerpiece. Rodrigo's Flowers, in contrast, charges $43 for the large centerpiece and $12 per arrangement for the rest. If Mrs. Williamson 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?
Let's call the number of small arrangements x.
At Silvergrove Florist, the cost of the large centerpiece is $29.
The cost of x small arrangements is $14 * x.
So the total cost at Silvergrove Florist would be $29 + $14 * x.
At Rodrigo's Flowers, the cost of the large centerpiece is $43.
The cost of x small arrangements is $12 * x.
So the total cost at Rodrigo's Flowers would be $43 + $12 * x.
Since the cost is the same at both flower shops, we can set up the equation:
$29 + $14 * x = $43 + $12 * x
To solve for x, we can subtract $12 * x from both sides:
$29 + $14 * x - $12 * x = $43 + $12 * x - $12 * x
$29 + $2 * x = $43
Next, we can subtract $29 from both sides:
$29 + $2 * x - $29 = $43 - $29
$2 * x = $14
Finally, we can divide both sides by $2:
$2 * x / $2 = $14 / $2
x = $7
Therefore, the total cost would be $29 + $14 * 7 = $29 + $98 = $127.
And Mrs. Williamson would need to order 7 small arrangements.