A designers spring line focuses on silk dresses, some with zippers an

Let's set up some equations to solve this problem. Let X represent the number of dresses with zippers and Y represent the number of dresses with buttons.

The total expense for fabric and notions can be calculated using the following equation:

Total Expense = Cost of Fabric + Cost of Zippers + Cost of Buttons

The cost of fabric is given as $28 per yard, and each dress requires 5 yards of silk, so the cost of fabric for X dresses with zippers and Y dresses with buttons is:

Cost of Fabric = 28 * 5 * (X + Y)

The cost of zippers is $2 each, and each dress with zippers requires one zipper. So, the cost of zippers for X dresses with zippers is:

Cost of Zippers = 2 * X

The cost of buttons is $1.50 each, and each dress with buttons requires 6 buttons. So, the cost of buttons for Y dresses with buttons is:

Cost of Buttons = 1.50 * 6 * Y

Given that the total expense for fabric and notions was $123,500, we can write the equation:

123,500 = 28 * 5 * (X + Y) + 2 * X + 1.50 * 6 * Y

Now, we can simplify this equation:

123,500 = 140 * (X + Y) + 2X + 9Y

123,500 = 140X + 140Y + 2X + 9Y

123,500 = 142X + 149Y

Now, we need to find the number of dresses with zippers (X) and the number of dresses with buttons (Y) that satisfy this equation.

To make this problem easier to solve, we can rearrange the equation to isolate one of the variables. Let's isolate X:

142X = 123,500 - 149Y

X = (123,500 - 149Y) / 142

Now, we need to find integer values for X and Y that make the equation true and non-negative. We'll start by trying different values for Y and see if we can find corresponding values for X:

Let Y = 0:
X = (123,500 - 149 * 0) / 142
X = 123,500 / 142 ≈ 869.01

Y = 0 doesn't give us an integer value for X, so let's try Y = 1:

X = (123,500 - 149 * 1) / 142
X = (123,500 - 149) / 142
X ≈ 869.46

Still not an integer. Let's try Y = 2:

X = (123,500 - 149 * 2) / 142
X = (123,500 - 298) / 142
X ≈ 868.75

Y = 3:

X = (123,500 - 149 * 3) / 142
X = (123,500 - 447) / 142
X ≈ 867.25

This process continues until we find integer values for X and Y that satisfy the equation. You can keep trying different values of Y until you find a combination that works. Keep in mind that both X and Y should be non-negative since you can't have a negative number of dresses.

Is there a shorter way of solving this apart from plugging in values?