x+2y+3z=14,x-y+2z =5, 2x + 2y +z= 11
1) x + 2y + 3z = 14
2) x - y + 2z = 5
3) 2x + 2y + z = 11
When there are three variables, it is easier to solve
by elimination, but apparently this is to be solved
by substitution.
I'll number the equations as I go along, so
you can see where I get each one I use later.
Solve equation 1 for x.
x + 2y + 3z = 14
4) x = -2y - 3z + 14
Substitute the value of x from equation 4 in place
of the x in equation 2.
x - y + 2z = 5
(-2y - 3z + 14) - y + 2z = 5
-2y - y - 3z + 2z + 14 = 5
-3y - z = 5 - 14
5) -3y - z = -9
Solve equation 5 for z.
-3y - z = -9
-z = 3y - 9
6) z = -3y + 9
Now, it gets tricky. Using equation 4 again, substitute the value
for x into equation 3.
2x + 2y + z = 11
2(-2y - 3z + 14) + 2y + z = 11
-4y - 6z + 28 + 2y + z = 11
-4y + 2y - 6z + z + 28 = 11
-2y - 5z + 28 = 11
-2y - 5z = 11 - 28
7) -2y - 5z = -17
Use the value of z from equation 6; substitute it into equation 7.
-2y - 5z = -17
-2y - 5(-3y + 9) = -17
-2y + 15y - 45 = -17
13y = -17 + 45
13y = 28
y = 28/13
Substitute that into equation 6 to solve for z.
z = -3y + 9
z = -3(28/13) + 9
z = -84/13 + 9
z = -84/13 + 117/13
z = 33/13
With values for y and z, we can solve for x. This
calls for equation 1.
x + 2y + 3z = 14
x + 2(28/13) + 3(33/13) = 14
x + 56/13 + 99/13 = 14
x + 155/13 = 14
x = 14 - 155/13
x = 182/13 - 155/13
x = 27/13
Check your work, using equation 2.
x - y + 2z = 5
27/13 - 28/13 + 2(33/13) = 5
-1/13 + 66/13 = 5
65/13 = 5
5 = 5
Also, use equation 3.
2x + 2y + z = 11
2(27/13) + 2(28/13) + 33/13 = 11
54/13 + 56/13 + 33/13 = 11
143/13 = 11
11 = 11
The computed values are correct.
Answer: x = 27/13, y = 28/13, z = 33/13