Three shrimp boats supply the shrimp wholesalers on Hilton Head with fresh catch. The Annabelle takes 50% of
its catch to Hudson's, 20% to Captain J's, and 30% to Mainstreet. The Curly Q takes 40% of its catch to Hudson's,
40% to Captain J's, and 20% to Mainstreet. The SloJoe takes 30% of its catch to Hudson's, 40% to Captain J's, and
30% to Mainstreet. One week Hudson's received 228.3 pounds of shrimp, Captain J's received 195.4 pounds, and
Mainstreet received 151.3 pounds. How many pounds of shrimp did each boat catch?
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To solve this problem, we can set up a system of equations to represent the information given. Let's represent the pounds of shrimp each boat catches as follows:
Let x be the pounds of shrimp caught by the Annabelle.
Let y be the pounds of shrimp caught by the Curly Q.
Let z be the pounds of shrimp caught by the SloJoe.
Based on the information given, we can write the following equations:
For Hudson's:
0.5x + 0.4y + 0.3z = 228.3
For Captain J's:
0.2x + 0.4y + 0.4z = 195.4
For Mainstreet:
0.3x + 0.2y + 0.3z = 151.3
These equations represent the distribution of the shrimp catch to the three wholesalers. Now we can solve this system of equations to find the values of x, y, and z.
There are several methods to solve a system of linear equations, such as substitution, elimination, or matrix methods. Let's use the substitution method:
We can start by solving one of the equations for one variable, and then substitute that into the other equations. Let's solve the first equation for x:
0.5x + 0.4y + 0.3z = 228.3
0.5x = 228.3 - 0.4y - 0.3z
x = (228.3 - 0.4y - 0.3z) / 0.5
Now we can substitute this expression for x into the other two equations:
0.2((228.3 - 0.4y - 0.3z) / 0.5) + 0.4y + 0.4z = 195.4
0.3((228.3 - 0.4y - 0.3z) / 0.5) + 0.2y + 0.3z = 151.3
Now we have two equations with only two variables, which we can solve using algebraic methods. We can simplify these equations to make them easier to work with.
Let's multiply all terms in the first equation by 0.5:
0.2(228.3 - 0.4y - 0.3z) + 0.4y + 0.4z = 97.7
Now, let's multiply all terms in the second equation by 0.5:
0.3(228.3 - 0.4y - 0.3z) + 0.2y + 0.3z = 75.65
Now we have two simplified equations:
0.2(228.3) - 0.08y - 0.06z + 0.4y + 0.4z = 97.7
0.3(228.3) - 0.12y - 0.09z + 0.2y + 0.3z = 75.65
Now we can simplify further:
45.66 - 0.08y - 0.06z + 0.4y + 0.4z = 97.7
68.49 - 0.12y - 0.09z + 0.2y + 0.3z = 75.65
Now let's combine like terms:
0.32y + 0.34z = 52.04
0.08y + 0.21z = 7.16
Now we have a new system of equations, which we can solve using substitution or elimination to find the values of y and z. I will use the elimination method to solve it from here.
Let's multiply the first equation by -0.08:
-0.026 y - 0.0272 z = -4.1616
0.08y + 0.21z = 7.16
Adding these two equations together:
-0.0272z + 0.21z = 2.9984
0.1828z = 2.9984
z ≈ 16.4
Now we can substitute this value of z back into one of the original equations to solve for y. I will use the second equation:
0.08y + 0.21z = 7.16
0.08y + 0.21(16.4) ≈ 7.16
0.08y + 3.444 ≈ 7.16
0.08y ≈ 7.16 - 3.444
0.08y ≈ 3.716
y ≈ 46.45
Finally, we can substitute the values of y and z back into one of the original equations to solve for x. I will use the first equation:
0.5x + 0.4y + 0.3z = 228.3
0.5x + 0.4(46.45) + 0.3(16.4) ≈ 228.3
0.5x + 18.58 + 4.92 ≈ 228.3
0.5x + 23.5 ≈ 228.3
0.5x ≈ 228.3 - 23.5
0.5x ≈ 204.8
x ≈ 409.6
Therefore, the Annabelle caught approximately 409.6 pounds of shrimp, the Curly Q caught approximately 46.45 pounds, and the SloJoe caught approximately 16.4 pounds.