Answer:
Amount of 3-inch bolts purchased = 18
Explanation:
We can determine the amount of 3-inch bolts James bought using a system of equations, where
- x represents the amount of 3-inch bolts purchased,
- and y represents the amount of 4-inch bolds purchased.
First equation:
We know that the sum of the amounts of 3-inch and 4-inch bolts equals the total amount of bolts purchased:
amount of 3-inch bolts + amount of 4-inch bolds = total amount of bolts
Since James bought 24 bolts, our first equation is given by:
x + y = 24
Second equation:
We know that the sum of the costs of the 3-inch and 4-inch bolts equals the total cost:
(price * amount of 3-inch bolts) + (price * amount of 4-inch bolts) = total cost
Since the 3-inch bolts cost $0.36 each, the 4-inch bolts cost $0.42 each, and the total cost was $9.00, our second equation is given by:
0.36x + 0.42y = 9
Method to solve: Substitution:
We can start by isolating y in the first equation:
(x + y = 24) - x
y = -x + 24
Solving for x (i.e., the number of 3-inch bolts purchased)
Now we can solve for x (i.e., the amount of 3-inch bolts purchased) by substituting y = -x + 24 in the second equation (0.36x + 0.42y = 9):
0.36x + 0.42(-x + 24) = 9
0.36x - 0.42x + 10.08 = 9
(-0.06x + 10.08 = 9) - 10.08
(-0.06x = -1.08) / -0.06
x = 18
Thus, the amount of 3-inch bolts purchased by James was 18.
Optional: Check the validity of the answer:
- In order to check that our answer is correct, we first need to find the amount of 4-inch bolts purchased (i.e., y).
We can do this by plugging in 18 for x in the first equation (x + y = 24):
(18 + y) = 24 - 18
y = 6
Since 18 + 6 = 24, we know the solutions hold true for the first equation.
Checking the solutions in the second equation (0.36x + 0.42y = 9):
Now we can check that the solutions hold true for the second equation by plugging in 18 for x and 6 for y in 0.36x + 0.42y = 9
0.36(18) + 0.42(6) = 9
6.48 + 2.52 = 9
9 = 9
Thus, the solutions are correct.