Answer:
x = 20/13 , y = 16/13 , z = 120/13
Explanation:
Solve the following system:
{x + y + z = 12 | (equation 1)
6 x - 2 y + z = 16 | (equation 2)
3 x + 4 y + 2 z = 28 | (equation 3)
Swap equation 1 with equation 2:
{6 x - 2 y + z = 16 | (equation 1)
x + y + z = 12 | (equation 2)
3 x + 4 y + 2 z = 28 | (equation 3)
Subtract 1/6 × (equation 1) from equation 2:
{6 x - 2 y + z = 16 | (equation 1)
0 x+(4 y)/3 + (5 z)/6 = 28/3 | (equation 2)
3 x + 4 y + 2 z = 28 | (equation 3)
Multiply equation 2 by 6:
{6 x - 2 y + z = 16 | (equation 1)
0 x+8 y + 5 z = 56 | (equation 2)
3 x + 4 y + 2 z = 28 | (equation 3)
Subtract 1/2 × (equation 1) from equation 3:
{6 x - 2 y + z = 16 | (equation 1)
0 x+8 y + 5 z = 56 | (equation 2)
0 x+5 y + (3 z)/2 = 20 | (equation 3)
Multiply equation 3 by 2:
{6 x - 2 y + z = 16 | (equation 1)
0 x+8 y + 5 z = 56 | (equation 2)
0 x+10 y + 3 z = 40 | (equation 3)
Swap equation 2 with equation 3:
{6 x - 2 y + z = 16 | (equation 1)
0 x+10 y + 3 z = 40 | (equation 2)
0 x+8 y + 5 z = 56 | (equation 3)
Subtract 4/5 × (equation 2) from equation 3:
{6 x - 2 y + z = 16 | (equation 1)
0 x+10 y + 3 z = 40 | (equation 2)
0 x+0 y+(13 z)/5 = 24 | (equation 3)
Multiply equation 3 by 5:
{6 x - 2 y + z = 16 | (equation 1)
0 x+10 y + 3 z = 40 | (equation 2)
0 x+0 y+13 z = 120 | (equation 3)
Divide equation 3 by 13:
{6 x - 2 y + z = 16 | (equation 1)
0 x+10 y + 3 z = 40 | (equation 2)
0 x+0 y+z = 120/13 | (equation 3)
Subtract 3 × (equation 3) from equation 2:
{6 x - 2 y + z = 16 | (equation 1)
0 x+10 y+0 z = 160/13 | (equation 2)
0 x+0 y+z = 120/13 | (equation 3)
Divide equation 2 by 10:
{6 x - 2 y + z = 16 | (equation 1)
0 x+y+0 z = 16/13 | (equation 2)
0 x+0 y+z = 120/13 | (equation 3)
Add 2 × (equation 2) to equation 1:
{6 x + 0 y+z = 240/13 | (equation 1)
0 x+y+0 z = 16/13 | (equation 2)
0 x+0 y+z = 120/13 | (equation 3)
Subtract equation 3 from equation 1:
{6 x+0 y+0 z = 120/13 | (equation 1)
0 x+y+0 z = 16/13 | (equation 2)
0 x+0 y+z = 120/13 | (equation 3)
Divide equation 1 by 6:
{x+0 y+0 z = 20/13 | (equation 1)
0 x+y+0 z = 16/13 | (equation 2)
0 x+0 y+z = 120/13 | (equation 3)
Collect results:
Answer: {x = 20/13 , y = 16/13 , z = 120/13