A hardware store makes a mixed bag of 42 items using screws, bolts, and washers. The cost of screws are $3.00 each, bolts cost $2.00 each, and washers are $1.50 each. The mixture calls for four times as many screws than bolts. The total cost of the mixture is $102.00. How much of each item did the store use?

s+b+w = 42

3s + 2b + 1.5w = 102.00
s = 4b

now crank it out.

s + b + w = 42

s = 4b

3s + 2b + 1.5w = 102

substituting
... 5b + w = 42
... 14b + 1.5w = 102

To solve this problem, we need to set up a system of equations based on the given information. Let's define the variables:

Let x represent the number of screws.
Let y represent the number of bolts.
Let z represent the number of washers.

Based on the information given, we can set up the following equations:

1) x + y + z = 42 (The total number of items in the mixed bag is 42)
2) 3x + 2y + 1.5z = 102 (The total cost of the mixture is $102.00)

Next, we are told that the mixture calls for four times as many screws as bolts. This can be expressed as:

3) x = 4y (The number of screws is four times the number of bolts)

Now we have a system of three equations:

x + y + z = 42
3x + 2y + 1.5z = 102
x = 4y

To solve this system of equations, we can use a method called substitution. We can solve equation 3) for x in terms of y: x = 4y. Then substitute this value into equations 1) and 2).

Substituting x = 4y into equation 1), we have:
4y + y + z = 42
5y + z = 42

Substituting x = 4y into equation 2), we have:
3(4y) + 2y + 1.5z = 102
12y + 2y + 1.5z = 102
14y + 1.5z = 102

Now we have a system of two equations with two variables:

5y + z = 42
14y + 1.5z = 102

We can solve this system by either substitution or elimination. Here, we will use the elimination method:

Step 1: Multiply the first equation by 1.5 to make the coefficients of z in both equations equal:
7.5y + 1.5z = 63

Step 2: Subtract the second equation from the modified first equation to eliminate z:
(7.5y + 1.5z) - (14y + 1.5z) = 63 - 102
7.5y - 14y = -39.00
-6.5y = -39.00

Step 3: Solve for y:
y = -39.00 / -6.5
y = 6

Now substitute the value of y into equation 1):

5(6) + z = 42
30 + z = 42
z = 42 - 30
z = 12

We have found the value of z.

Next, substitute the values of y and z into equation 3):

x = 4(6)
x = 24

We have found the values of x, y, and z.

Therefore, the hardware store used 24 screws, 6 bolts, and 12 washers in the mixed bag.