Final answer:
To solve the system of equations, we can use the method of elimination or substitution. By multiplying and subtracting the equations, we eliminate variables to determine their values. However, in this case, we find that the system is inconsistent or contradictory, meaning there is no solution.
Step-by-step explanation:
To solve the system of equations:
3x + 5y - 2z = 0
7x - 4y + 8z = 37
-4x - 9y + z = 7
We can use the method of elimination or substitution to find the values of x, y, and z.
Let's use the elimination method:
Multiply the first equation by 7, the second equation by 3, and the third equation by 4 to make the coefficients of x equal in the first two equations:
21x + 35y - 14z = 0
21x - 12y + 24z = 111
-16x - 36y + 4z = 28
Subtract the second equation from the first equation:
(21x + 35y -14z) - (21x - 12y + 24z) = 0 - 111
-47y - 38z = -111
Subtract twice the second equation from the third equation:
(-16x - 36y + 4z) - 2(21x - 12y + 24z) = 28 - 2(111)
13x - 60y - 44z = -194
Now we have two equations:
-47y - 38z = -111
13x - 60y - 44z = -194
Let's multiply the first equation by 13 and the second equation by -47 to make the coefficients of y equal:
-611y - 494z = -1443
-611y + 2820y + 2088z = 9121
Add the two equations:
(-611y - 494z) + (-611y + 2820y + 2088z) = -1443 + 9121
2326y + 1594z = 7678
Now we have two equations:
2326y + 1594z = 7678
13x - 60y - 44z = -194
Multiply the first equation by 44 and the second equation by 1594 to make the coefficients of z equal:
102344y + 70236z = 338712
20702x - 95640y - 70256z = -31076
Add the two equations:
(102344y + 70236z) + (20702x - 95640y - 70256z) = 338712 - 31076
20702x - 25404y = 307636
Now we have two equations:
20702x - 25404y = 307636
13x - 60y = -194
Multiply the second equation by -1692:
-1692(13x - 60y) = -1692(-194)
-1692(13x) - 1692(-60y) = 333768
-21996x + 101520y = 333768
Add the two equations:
(20702x - 25404y) + (-21996x + 101520y) = 307636 + 333768
-1294x + 76116y = 641404
Now we have two equations:
-1294x + 76116y = 641404
20702x - 25404y = 307636
Multiply the first equation by 16:
16(-1294x + 76116y) = 16(641404)
-20704x + 1217856y = 10262544
Add the two equations:
(-20704x + 1217856y) + (20702x - 25404y) = 10262544 + 307636
-2x + 1192452y = 10570180
Now we have two equations:
-2x + 1192452y = 10570180
-2x + 596226y = 5285090
Subtract the second equation from the first equation:
(-2x + 1192452y) - (-2x + 596226y) = 10570180 - 5285090
596226y - 596226y = 5285090
0 = 5285090
This equation is not possible, which means there is no solution to the system of equations. The system is inconsistent or contradictory.