Final answer:
To solve the given system of equations using Gaussian elimination, follow these steps: multiply equations to match coefficients, add or subtract equations, isolate variables, and solve for each variable one by one. The solution to the system of equations is (d, e, f) = (-18.43, -6.259, 2.446).
Step-by-step explanation:
To solve the given system of equations using Gaussian elimination, we will eliminate variables one by one to find the values of d, e, and f. Here are the steps:
Step 1: Multiply the second equation by 3 and the third equation by 2 to make the coefficients of d in the second and third equations match:
-6d + 3e + 144f = 819
8d + 20e - 40f = -300
Step 2: Add the first equation to the second equation and the third equation to eliminate d:
-6d + 3e + 144f = 819
8d + 20e - 40f = -300
-------------------------
23e + 104f = 519
Step 3: Multiply the first equation by 2 and the third equation by 3 to make the coefficients of d in the first and third equations match:
12d + 36e - 36f = -336
Step 4: Add the first equation to the third equation to eliminate d:
12d + 36e - 36f = -336
-------------------------
36e - 36f = -336
Step 5: Now, we have the following system of equations:
23e + 104f = 519
36e - 36f = -336
Step 6: Multiply the second equation by 23 and the first equation by 36 to make the coefficients of e in the two equations match:
828e - 828f = -7560
828e + 3744f = 18744
Step 7: Add the two equations to eliminate e:
828e - 828f = -7560
828e + 3744f = 18744
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4572f = 11184
Step 8: Solve for f:
f = 11184/4572 = 2.446
Step 9: Substitute the value of f back into the second equation to solve for e:
36e - 36(2.446) = -336
Solving for e, we find e = -6.259
Step 10: Substitute the values of e and f back into the first equation to solve for d:
-6d + 3(-6.259) + 144(2.446) = 819
Solving for d, we find d = -18.43
Therefore, the solution to the system of equations is (d, e, f) = (-18.43, -6.259, 2.446).