2x+y+3z=-12
3x-2y-z=3
-x+5y+2z=-3
I did it and I got (-1/3, 14/5, 212/15) and I think I did something wrong. Could someone please do this and tell me what you got? (I know how to do it, I just want to know what people got)
Solve the following system:
{2 x + y + 3 z = -12 | (equation 1)
3 x - 2 y - z = 3 | (equation 2)
-x + 5 y + 2 z = -3 | (equation 3)
Swap equation 1 with equation 2:
{3 x - 2 y - z = 3 | (equation 1)
2 x + y + 3 z = -12 | (equation 2)
-x + 5 y + 2 z = -3 | (equation 3)
Subtract 2/3 × (equation 1) from equation 2:
{3 x - 2 y - z = 3 | (equation 1)
0 x+(7 y)/3 + (11 z)/3 = -14 | (equation 2)
-x + 5 y + 2 z = -3 | (equation 3)
Multiply equation 2 by 3:
{3 x - 2 y - z = 3 | (equation 1)
0 x+7 y + 11 z = -42 | (equation 2)
-x + 5 y + 2 z = -3 | (equation 3)
Add 1/3 × (equation 1) to equation 3:
{3 x - 2 y - z = 3 | (equation 1)
0 x+7 y + 11 z = -42 | (equation 2)
0 x+(13 y)/3 + (5 z)/3 = -2 | (equation 3)
Multiply equation 3 by 3:
{3 x - 2 y - z = 3 | (equation 1)
0 x+7 y + 11 z = -42 | (equation 2)
0 x+13 y + 5 z = -6 | (equation 3)
Swap equation 2 with equation 3:
{3 x - 2 y - z = 3 | (equation 1)
0 x+13 y + 5 z = -6 | (equation 2)
0 x+7 y + 11 z = -42 | (equation 3)
Subtract 7/13 × (equation 2) from equation 3:
{3 x - 2 y - z = 3 | (equation 1)
0 x+13 y + 5 z = -6 | (equation 2)
0 x+0 y+(108 z)/13 = (-504)/13 | (equation 3)
Multiply equation 3 by 13/36:
{3 x - 2 y - z = 3 | (equation 1)
0 x+13 y + 5 z = -6 | (equation 2)
0 x+0 y+3 z = -14 | (equation 3)
Divide equation 3 by 3:
{3 x - 2 y - z = 3 | (equation 1)
0 x+13 y + 5 z = -6 | (equation 2)
0 x+0 y+z = (-14)/3 | (equation 3)
Subtract 5 × (equation 3) from equation 2:
{3 x - 2 y - z = 3 | (equation 1)
0 x+13 y+0 z = 52/3 | (equation 2)
0 x+0 y+z = -14/3 | (equation 3)
Divide equation 2 by 13:
{3 x - 2 y - z = 3 | (equation 1)
0 x+y+0 z = 4/3 | (equation 2)
0 x+0 y+z = -14/3 | (equation 3)
Add 2 × (equation 2) to equation 1:
{3 x + 0 y - z = 17/3 | (equation 1)
0 x+y+0 z = 4/3 | (equation 2)
0 x+0 y+z = -14/3 | (equation 3)
Add equation 3 to equation 1:
{3 x+0 y+0 z = 1 | (equation 1)
0 x+y+0 z = 4/3 | (equation 2)
0 x+0 y+z = -14/3 | (equation 3)
Divide equation 1 by 3:
{x+0 y+0 z = 1/3 | (equation 1)
0 x+y+0 z = 4/3 | (equation 2)
0 x+0 y+z = -14/3 | (equation 3)
Collect results:
Answer: |x = 1/3 y = 4/3 z = -14/3
+129839 2x+y+3z = -12 (1)
3x-2y-z = 3 (2)
-x+5y+2z = -3 (3)
Multiply equation 3 by 2 add to equation 1
11y + 7z = -18 (4)
Multiply equation 3 by 3 add to equation 2
13y + 5z = -6 (5)
Multiply equation 4 by -5
-55y - 35z = 90 (6)
Multiply equation 5 by 7
91y + 35z = -42 (7)
Add (6) and (7)
36y = 48 → y = 48/36 = 4/3
And using (4) .....11(4/3) + 7z = -18 → -54/3 - 44/3 = 7z → z = -98/21 = - 14/3
And using (3) .....-x+5(4/3)+2(-14/3) = -3 → x = 20/3 - 28/3 + 9/3 = 1/3
So {x, y, z } = { 1/3, 4/3, - 14/3 }
