x = -1/5
, y = 4/5
, z = 3/5
Step-by-step explanation:
Pls note a typo in your question 2x+5y=6z=6, I changed it to 2x+5y=6z, in any case if you didn't make a typo there is no solution.
Solve the following system:
{2 x + 2 y + 3 z = 3 | (equation 1)
{6 x + 3 y + 3 z = 3 | (equation 2)
{2 x + 5 y = 6 z | (equation 3)
Express the system in standard form:
{2 x + 2 y + 3 z = 3 | (equation 1)
{6 x + 3 y + 3 z = 3 | (equation 2)
{2 x + 5 y - 6 z = 0 | (equation 3)
Swap equation 1 with equation 2:
{6 x + 3 y + 3 z = 3 | (equation 1)
{2 x + 2 y + 3 z = 3 | (equation 2)
{2 x + 5 y - 6 z = 0 | (equation 3)
Subtract 1/3 × (equation 1) from equation 2:
{6 x + 3 y + 3 z = 3 | (equation 1)
{0 x+y + 2 z = 2 | (equation 2)
{2 x + 5 y - 6 z = 0 | (equation 3)
Divide equation 1 by 3:
{2 x + y + z = 1 | (equation 1)
{0 x+y + 2 z = 2 | (equation 2)
{2 x + 5 y - 6 z = 0 | (equation 3)
Subtract equation 1 from equation 3:
{2 x + y + z = 1 | (equation 1)
{0 x+y + 2 z = 2 | (equation 2)
{0 x+4 y - 7 z = -1 | (equation 3)
Swap equation 2 with equation 3:
{2 x + y + z = 1 | (equation 1)
{0 x+4 y - 7 z = -1 | (equation 2)
{0 x+y + 2 z = 2 | (equation 3)
Subtract 1/4 × (equation 2) from equation 3:
{2 x + y + z = 1 | (equation 1)
{0 x+4 y - 7 z = -1 | (equation 2)
{0 x+0 y+(15 z)/4 = 9/4 | (equation 3)
Multiply equation 3 by 4/3:
{2 x + y + z = 1 | (equation 1)
{0 x+4 y - 7 z = -1 | (equation 2)
{0 x+0 y+5 z = 3 | (equation 3)
Divide equation 3 by 5:
{2 x + y + z = 1 | (equation 1)
{0 x+4 y - 7 z = -1 | (equation 2)
{0 x+0 y+z = 3/5 | (equation 3)
Add 7 × (equation 3) to equation 2:
{2 x + y + z = 1 | (equation 1)
{0 x+4 y+0 z = 16/5 | (equation 2)
{0 x+0 y+z = 3/5 | (equation 3)
Divide equation 2 by 4:
{2 x + y + z = 1 | (equation 1)
{0 x+y+0 z = 4/5 | (equation 2)
{0 x+0 y+z = 3/5 | (equation 3)
Subtract equation 2 from equation 1:
{2 x + 0 y+z = 1/5 | (equation 1)
{0 x+y+0 z = 4/5 | (equation 2)
{0 x+0 y+z = 3/5 | (equation 3)
Subtract equation 3 from equation 1:
{2 x+0 y+0 z = (-2)/5 | (equation 1)
{0 x+y+0 z = 4/5 | (equation 2)
{0 x+0 y+z = 3/5 | (equation 3)
Divide equation 1 by 2:
{x+0 y+0 z = (-1)/5 | (equation 1)
{0 x+y+0 z = 4/5 | (equation 2)
{0 x+0 y+z = 3/5 | (equation 3)
Collect results:
 {x = -1/5
, y = 4/5
, z = 3/5
