I want to know how to do this algebraically. Not with a matrix. There are 5 equations with 5 variables. I put the 0's out there to help organize. I need to solve for all of them. Please help!
1A + 2B + 3C + 2D + 0E = 19.968
1A + 1B + 4C + 2D + 0E = 18.818
1A + 4B + 0C + 3D + 0E = 24.592
0A + 3B + 1C + 2D + 1E = 22.312
0A + 0B + 0C + 2D + 1E = 15.882
1A + 2B + 3C + 2D + 0E = 19.968
1A + 1B + 4C + 2D + 0E = 18.818
1A + 4B + 0C + 3D + 0E = 24.592
0A + 3B + 1C + 2D + 1E = 22.312
0A + 0B + 0C + 2D + 1E = 15.882
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You don't have to do it this way, but to make things easier, start by making sure that the first equation has A, the second has B, the third has C, the fourth has D and the fifth has E -- easily accomplished by reversing the third and fourth equation above.
Next, solve equation 1 for A and plug that into equation 2 for A. Then solve equation 2 for B and plug that into equation 3. Then solve equation 3 for C and plug that into equation 4. Then solve equation 4 for D and plug that into equation 5. Now solve equation 5 for E. Knowing E, go back to your revised equation 4 and solve for D. Knowing D, go back to your revised euation 3 and solve for C. Keep working backwards until you have the answer for all 5 variables.
I'll start you out, then you follow the sequence described above and do the rest.
A + 2B + 3C + 2D = 19968
A = 19968 - 2B - 3C - 2D
A + B + 4C + 2D = 18818
Substitute 19968 - 2B - 3C - 2D for A.
19968 - 2B - 3C - 2D + B + 4C + 2D = 18818
Solve for B, then substitute that into the nex equation.
Next, solve equation 1 for A and plug that into equation 2 for A. Then solve equation 2 for B and plug that into equation 3. Then solve equation 3 for C and plug that into equation 4. Then solve equation 4 for D and plug that into equation 5. Now solve equation 5 for E. Knowing E, go back to your revised equation 4 and solve for D. Knowing D, go back to your revised euation 3 and solve for C. Keep working backwards until you have the answer for all 5 variables.
I'll start you out, then you follow the sequence described above and do the rest.
A + 2B + 3C + 2D = 19968
A = 19968 - 2B - 3C - 2D
A + B + 4C + 2D = 18818
Substitute 19968 - 2B - 3C - 2D for A.
19968 - 2B - 3C - 2D + B + 4C + 2D = 18818
Solve for B, then substitute that into the nex equation.
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Gauss-Jordan Elimination.
Use equation 1 to eliminate the A terms from equations 2 and 3. The system becomes
A+2B+3C+2D = 19.968
-B+C = -1.15
2B-3C+D = 4.624
3B+C+2D+E = 22.312
2D+E = 15.882
Divide equation 2 by -1:
B-C = 1.15
Use equation 2 to eliminate the B terms from equations 1, 3, 4, and 5. The system becomes
A+5C+2D = 17.668
B-C = 1.15
-C+D = 2.324
4C+2D+E = 18.862
2D+E = 15.882
Divide equation 3 by -1:
C-D = -2.324
Use equation 3 to eliminate the C terms from equations 1, 2, 4, and 5. The system becomes
A7D = 29.288
B-D = -1.174
C-D = -2.324
6D+E = 28.158
2D+E = 15.882
Divide equation 4 by 6:
D+(1/6)E = 4.693
Use equation 4 to eliminate the D terms from equations 1, 2, 3, and 5. The system becomes
A - (7/6)E = -3.563
B + (1/6)E = 3.519
C + (1/6)E = 2.369
D + (1/6)E = 4.693
(2/3)E = 6.496
Divide equation 5 by 2/3:
E = 9.744
Use equation 5 to eliminate the E terms from equations 1, 2, 3, and 4. The system becomes
A = 7.805
B = 1.895
C = 0.745
D = 3.069
E = 9.744
Use equation 1 to eliminate the A terms from equations 2 and 3. The system becomes
A+2B+3C+2D = 19.968
-B+C = -1.15
2B-3C+D = 4.624
3B+C+2D+E = 22.312
2D+E = 15.882
Divide equation 2 by -1:
B-C = 1.15
Use equation 2 to eliminate the B terms from equations 1, 3, 4, and 5. The system becomes
A+5C+2D = 17.668
B-C = 1.15
-C+D = 2.324
4C+2D+E = 18.862
2D+E = 15.882
Divide equation 3 by -1:
C-D = -2.324
Use equation 3 to eliminate the C terms from equations 1, 2, 4, and 5. The system becomes
A7D = 29.288
B-D = -1.174
C-D = -2.324
6D+E = 28.158
2D+E = 15.882
Divide equation 4 by 6:
D+(1/6)E = 4.693
Use equation 4 to eliminate the D terms from equations 1, 2, 3, and 5. The system becomes
A - (7/6)E = -3.563
B + (1/6)E = 3.519
C + (1/6)E = 2.369
D + (1/6)E = 4.693
(2/3)E = 6.496
Divide equation 5 by 2/3:
E = 9.744
Use equation 5 to eliminate the E terms from equations 1, 2, 3, and 4. The system becomes
A = 7.805
B = 1.895
C = 0.745
D = 3.069
E = 9.744