solve for x y z 3x-2y+2z=1 2x+5y-5z=7 4x-3y+z=-3 by substitution and elimination
solve algebraically
1) 3x - 2y + 2z = 1
2) 2x + 5y - 5z = 7
3) 4x - 3y + z = -3
Multiply equation 1 by 5.
5(3x - 2y + 2z) = 1 * 5
4) 15x - 10y + 10z = 5
Multiply equation 2 by 2.
2(2x + 5y - 5z) = 7 * 2
5) 4x + 10y - 10z = 14
Add equation 5 to equation 4.
15x - 10y + 10z = 5
+(4x + 10y - 10z = 14)
----------------------------
19x = 19
6) 19x = 19
x = 1 <<<<<<<<<<<<<<<<<
Multiply equation 3 by 5.
5(4x - 3y + z) = -3 * 5
7) 20x - 15y + 5z = -15
Add equation 7 to equation 2.
2x + 5y - 5z = 7
+(20x - 15y + 5z = -15)
------------------------------
22x - 10y = -8
8) 22x - 10y = -8
Solve equation 8 for y.
22x - 10y = -8
22(1) - 10y = -8
22 - 10y = -8
-10y = -8 - 22
-10y = -30
y = 3 <<<<<<<<<<<<<<<<<
Use equation 3 to solve for z.
4x - 3y + z = -3
4(1) - 3(3) + z = -3
4 - 9 + z = -3
-5 + z = -3
z = -3 + 5
z = 2 <<<<<<<<<<<<<<<<<
Check, check, check. Using the original equations, of course.
1) 3x - 2y + 2z = 1
3(1) - 2(3) + 2(2) = 1
3 - 6 + 4 = 1
1 = 1
2) 2x + 5y - 5z = 7
2(1) + 5(3) - 5(2) = 7
2 + 15 - 10 = 7
7 = 7
3) 4x - 3y + z = -3
4(1) - 3(3) + (2) = -3
4 - 9 + 2 = -3
-3 = -3
Correct.
Answer: x = 1, y = 3, z = 2