Answer:
x = 10
, y = -7
, z = 1
Step-by-step explanation:
Solve the following system:
{x - 3 z = 7 | (equation 1)
2 x + y - 2 z = 11 | (equation 2)
-x - 2 y + 9 z = 13 | (equation 3)
Swap equation 1 with equation 2:
{2 x + y - 2 z = 11 | (equation 1)
x + 0 y - 3 z = 7 | (equation 2)
-x - 2 y + 9 z = 13 | (equation 3)
Subtract 1/2 × (equation 1) from equation 2:
{2 x + y - 2 z = 11 | (equation 1)
0 x - y/2 - 2 z = 3/2 | (equation 2)
-x - 2 y + 9 z = 13 | (equation 3)
Multiply equation 2 by 2:
{2 x + y - 2 z = 11 | (equation 1)
0 x - y - 4 z = 3 | (equation 2)
-x - 2 y + 9 z = 13 | (equation 3)
Add 1/2 × (equation 1) to equation 3:
{2 x + y - 2 z = 11 | (equation 1)
0 x - y - 4 z = 3 | (equation 2)
0 x - (3 y)/2 + 8 z = 37/2 | (equation 3)
Multiply equation 3 by 2:
{2 x + y - 2 z = 11 | (equation 1)
0 x - y - 4 z = 3 | (equation 2)
0 x - 3 y + 16 z = 37 | (equation 3)
Swap equation 2 with equation 3:
{2 x + y - 2 z = 11 | (equation 1)
0 x - 3 y + 16 z = 37 | (equation 2)
0 x - y - 4 z = 3 | (equation 3)
Subtract 1/3 × (equation 2) from equation 3:
{2 x + y - 2 z = 11 | (equation 1)
0 x - 3 y + 16 z = 37 | (equation 2)
0 x+0 y - (28 z)/3 = (-28)/3 | (equation 3)
Multiply equation 3 by -3/28:
{2 x + y - 2 z = 11 | (equation 1)
0 x - 3 y + 16 z = 37 | (equation 2)
0 x+0 y+z = 1 | (equation 3)
Subtract 16 × (equation 3) from equation 2:
{2 x + y - 2 z = 11 | (equation 1)
0 x - 3 y+0 z = 21 | (equation 2)
0 x+0 y+z = 1 | (equation 3)
Divide equation 2 by -3:
{2 x + y - 2 z = 11 | (equation 1)
0 x+y+0 z = -7 | (equation 2)
0 x+0 y+z = 1 | (equation 3)
Subtract equation 2 from equation 1:
{2 x + 0 y - 2 z = 18 | (equation 1)
0 x+y+0 z = -7 | (equation 2)
0 x+0 y+z = 1 | (equation 3)
Add 2 × (equation 3) to equation 1:
{2 x+0 y+0 z = 20 | (equation 1)
0 x+y+0 z = -7 | (equation 2)
0 x+0 y+z = 1 | (equation 3)
Divide equation 1 by 2:
{x+0 y+0 z = 10 | (equation 1)
0 x+y+0 z = -7 | (equation 2)
0 x+0 y+z = 1 | (equation 3)
Collect results:
Answer: {x = 10
, y = -7
, z = 1
is automatically rational because it can be expressed as a fraction. Another way to find out if it's rational is to convert it to a decimal, then see if the decimal is terminating or repeating.