SOLUTION: SYSTEM OF 3 EQUATIONS WITH 3 UNKNOWNS FIND SOLUTION SET FOR THE FOLLOWING SYSTEM OF EQUATIONS: X -Y +5Z = 2 4X -3Y +5Z = 3 3X -2Y +4Z = 1

Algebra ->  Coordinate Systems and Linear Equations  -> Lessons -> SOLUTION: SYSTEM OF 3 EQUATIONS WITH 3 UNKNOWNS FIND SOLUTION SET FOR THE FOLLOWING SYSTEM OF EQUATIONS: X -Y +5Z = 2 4X -3Y +5Z = 3 3X -2Y +4Z = 1       Log On


   

Question 208209: SYSTEM OF 3 EQUATIONS WITH 3 UNKNOWNS

FIND SOLUTION SET FOR THE FOLLOWING SYSTEM OF EQUATIONS:
X -Y +5Z = 2
4X -3Y +5Z = 3
3X -2Y +4Z = 1
Found 2 solutions by pjalgeb, Theo:
Answer by pjalgeb(9) About Me  (Show Source):
You can put this solution on YOUR website!

SOLUTION:
The augmented 3x3 matrix for the above system of equations is:
| 1 -1 5 | 2 |
| 4 -3 5 | 3 |
| 3 -2 4 | 1 |

By Cramers Rule:
x+=+Dx%2FD
y+=+Dy%2FD
z+=+Dz%2FD

First, lets solve for D, Dx, Dy and Dz:

D = | 1 -1 5 |
| 4 -3 5 |
| 3 -2 4 |
D = 1(-12+10) - 4(-4+10) + 3(-5+15)
D = 1(-2) -4(6) + 3(10)
D = -2 -24 +30
D = -26 +30
D+=+4

Dx = | 2 -1 5 |
| 3 -3 5 |
| 1 -2 4 |
Dx = 2(-12+10) - 3(-4+10) +1(-5+15)
Dx = 2(-2) - 3(6) + 1(10)
Dx = -4 - 18 + 10
Dx = -22 +10
Dx+=+-12

Dy = | 1 2 5 |
| 4 3 5 |
| 3 1 4 |
Dy = 1(12-5) -4(8-5) +3(10-15)
Dy = 1(7) -4(3) +3(-5)
Dy = 7 -12 -15
Dy = -5 -15
Dy+=+-20

Dz = | 1 -1 2 |
| 4 -3 3 |
| 3 -2 1 |
Dz = 1(-3+6)- 4(-1+4)+ 3(-3+6)
Dz = 1(3) - 4(3) + 3(3)
Dz = 3 -12 + 9
Dz = -9 + 9
Dz+=+0


x+=+Dx%2FD+=+-12%2F4
x+=+-3
y+=+Dy%2FD+=+-20%2F4
y+=+-5
z+=+Dz%2FD+=+0%2F4
z+=+0

Answer: The solution set is {(-3,-5, 0)}



Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
you have 3 equations in 3 unknowns.
you want to reduce that to 2 equations in 2 unknowns
you then reduce that to 1 equation in one unknown
working your way back up the chain...
you then solve for the first unknown
you then solve for the second unknown
you then solve for the third unknown.
-----
your equations are:
x - y + 5z = 2 = equation 1
4x - 3y + 5z = 3 = equation 2
3x - 2y + 4z = 1 = equation 3
-----
you can eliminate z first.
-----
you subtract equation 1 from equation 2 to get:
3x - 2y = 1 = equation 4
-----
you multiply equation 1 by 4 to get:
4x - 4y + 20z = 8 = equation 5
-----
you multiply equation 3 by 5 to get:
15x - 10y + 20z = 5 = equation 6
-----
you subtract equation 5 from equation 6 to get:
11x - 6y = -3 = equation 7
-----
you now have 2 equations in 2 unknowns.
those are equation 4 and equation 7 shown below:
3x - 2y = 1 = equation 4
11x - 6y = -3 = equation 7
-----
next you will eliminate the y to get one equation in one unknown.,
-----
you multiply equation 4 by 3 to get:
9x - 6y = 3 = equation 8
-----
you subtract equation 8 from equation 7 to get:
2x = -6
-----
you now have one equation in one unknown which you solve to get:
x = -3
-----
you now go back to equation 4 or equation 7 and solve for y.
let's use equation 7
-----
11x - 6y = -3 = equation 7
replace x with -3 to get:
11 *(-3) - 6y = -3
-33 - 6y = -3
6y = -30
y = -5
-----
you now have:
x = -3
y = -5
-----
go back to equation 1 or 2 or 3 and replace x with -3 and replace y with -5 to solve for z.
we'll use equation 2 as shown below:
-----
4x - 3y + 5z = 3 = equation 2
replace x with -3 and y with -5 to get:
4 * (-3) - 3 * (-5) + 5z = 3
-12 + 15 + 5z = 3
3 + 5z = 3
5z = 0
z = 0
-----
you now have:
x = -3
y = -5
z = 0
-----
your equation should be solved and you confirm that by replacing x with -3 and y with -5 and z with 0 in all 3 of your equations to make sure that the equations are true with those values in them for x,y,z.
-----
your original equations are:
x - y + 5z = 2 = equation 1
4x - 3y + 5z = 3 = equation 2
3x - 2y + 4z = 1 = equation 3
-----
equation 1 becomes -3 + 5 + 0 = 2 which becomes 2 = 2 which is true.
equation 2 becomes -12 + 15 + 0 = 3 which becomes 3 = 3 which is true.
equation 3 becomes -9 + 10 + 0 = 1 which becomes 1 = 3 which is true.
-----
all 3 equations are true so the values for x,y,z satisfy all of them simultaneously and you are done.
-----