a sistem equation
3x-5y-2z=-5
6x+2y+3z=5
4x-3y-z=8
x=? ;y=? ; z=?
1) 3x - 5y - 2z = -5
2) 6x + 2y + 3z = 5
3) 4x - 3y - z = 8
The way to solve systems like this is to sequentially eliminate
each unknown so that there is only one left. That lets you solve
for the value of that remaining unknown, and substitute that
into one of the equations, then eliminate one of the other two
unknowns. You then solve for the value of the second unknown.
Now, you substitute both of those into one of the equations
to solve for the third unknown.
We'll use the first and second equation to eliminate the x term.
Multiply equation 1 by 2.
2 * (3x - 5y - 2z) = -5 * 2
4) 6x - 10y - 4z = -10
Subtract equation 2 from equation 4.
6x - 10y - 4z = -10
-(6x + 2y + 3z = 5)
-------------------------
- 12y - 7z = -15
5) -12y - 7z = -15
That gives us one equation with y and z. Use equations
2 and 3 to eliminate x again, giving us a second equation
with y and z.
We need to multiply equation 2 by 2, making 12 the co-efficient of x.
Also, multiply equation 3 by 3, again making 12 the co-efficient of x.
2 * (6x + 2y + 3z) = 5 * 2
6) 12x + 4y + 6z = 10
3 * (4x - 3y - z) = 8 * 3
7) 12x - 9y - 3z = 24
Subtract equation 6 from equation 7.
12x - 9y - 3z = 24
-(12x + 4y + 6z = 10)
--------------------------
- 13y - 9z = 14
8) -13y - 9z = 14
Using equations 5 and 8, we will eliminate the z term.
Multiply equation 5 by 9.
9 * (-12y - 7z) = -15 * 9
9) -108y - 63z = -135
Multiply equation 8 by 7.
7 * (-13y - 9z) = 14 * 7
10) -91y - 63z = 98
Subtract equation 10 from equation 9.
-108y - 63z = -135
-(-91y - 63z = 98)
------------------------
-17y = -233
y = 13.70588 <<<<<<<<<<<<<<<<<<<<
Use that in equation 10 to solve for z.
-91y - 63z = 98
-91(13.70588) - 63z = 98
-1247.23508 - 63z = 98
-63z = 98 + 1247.23508
-63z = 1345.23508
z = 1666/3367 / -63
z = -21.35293 <<<<<<<<<<<<<<<<<<<<
Use equation 1 to solve for x.
3x - 5y - 2z = -5
3x - 5(13.70588) - 2(-21.35293) = -5
x = 6.94118 <<<<<<<<<<<<<<<<<<<<
Use equation 2 to check the values.
6x + 2y + 3z = 5
6(6.94118) + 2(13.70588) + 3(-21.35293) = 5
41.64708 + 27.41176 - 64.05879 = 5
5.00005 = 5
Rounding errors account for the discrepancy.
Use equation 3 to check the values.
4x - 3y - z = 8
4(6.94118) - 3(13.70588) - (-21.35293) = 8
27.76472 - 41.11764 + 21.35293 = 8
8.00001 = 8
Again, consider rounding errors.
x = 6.94118; y = 13.70588; z = -21.35293