Problem to solve
Solve \[\frac{x}{3}-\frac{y}{2}+z=2\], \[-x-\frac{y}{3}+2z=3\], \[2x-3y+\frac{z}{3}=-1\]
Solution
STEP 1
Solve the following system:
{x/3 - y/2 + z = 2 | (equation 1)
-x - y/3 + 2 z = 3 | (equation 2)
2 x - 3 y + z/3 = -1 | (equation 3)
STEP 2
Swap equation 1 with equation 3:
{2 x - 3 y + z/3 = -1 | (equation 1)
-x - y/3 + 2 z = 3 | (equation 2)
x/3 - y/2 + z = 2 | (equation 3)
STEP 3
Add 1/2 × (equation 1) to equation 2:
{2 x - 3 y + z/3 = -1 | (equation 1)
0 x - (11 y)/6 + (13 z)/6 = 5/2 | (equation 2)
x/3 - y/2 + z = 2 | (equation 3)
STEP 4
Multiply equation 1 by 3:
{6 x - 9 y + z = -3 | (equation 1)
0 x - (11 y)/6 + (13 z)/6 = 5/2 | (equation 2)
x/3 - y/2 + z = 2 | (equation 3)
STEP 5
Multiply equation 2 by 6:
{6 x - 9 y + z = -3 | (equation 1)
0 x - 11 y + 13 z = 15 | (equation 2)
x/3 - y/2 + z = 2 | (equation 3)
STEP 6
Multiply equation 3 by 6:
{6 x - 9 y + z = -3 | (equation 1)
0 x - 11 y + 13 z = 15 | (equation 2)
2 x - 3 y + 6 z = 12 | (equation 3)
STEP 7
Subtract 1/3 × (equation 1) from equation 3:
{6 x - 9 y + z = -3 | (equation 1)
0 x - 11 y + 13 z = 15 | (equation 2)
0 x+0 y+(17 z)/3 = 13 | (equation 3)
STEP 8
Multiply equation 3 by 3:
{6 x - 9 y + z = -3 | (equation 1)
0 x - 11 y + 13 z = 15 | (equation 2)
0 x+0 y+17 z = 39 | (equation 3)
STEP 9
Divide equation 3 by 17:
{6 x - 9 y + z = -3 | (equation 1)
0 x - 11 y + 13 z = 15 | (equation 2)
0 x+0 y+z = 39/17 | (equation 3)
STEP 10
Subtract 13 × (equation 3) from equation 2:
{6 x - 9 y + z = -3 | (equation 1)
0 x - 11 y+0 z = -252/17 | (equation 2)
0 x+0 y+z = 39/17 | (equation 3)
STEP 11
Divide equation 2 by -11:
{6 x - 9 y + z = -3 | (equation 1)
0 x+y+0 z = 252/187 | (equation 2)
0 x+0 y+z = 39/17 | (equation 3)
STEP 12
Add 9 × (equation 2) to equation 1:
{6 x + 0 y+z = 1707/187 | (equation 1)
0 x+y+0 z = 252/187 | (equation 2)
0 x+0 y+z = 39/17 | (equation 3)
STEP 13
Subtract equation 3 from equation 1:
{6 x+0 y+0 z = 1278/187 | (equation 1)
0 x+y+0 z = 252/187 | (equation 2)
0 x+0 y+z = 39/17 | (equation 3)
STEP 14
Divide equation 1 by 6:
{x+0 y+0 z = 213/187 | (equation 1)
0 x+y+0 z = 252/187 | (equation 2)
0 x+0 y+z = 39/17 | (equation 3)
STEP 15
Collect results:
Answer: | 
 | {x = 213/187
y = 252/187
z = 39/17