To solve this system of equations, we can use the method of elimination or substitution. I will use the method of elimination.
Step 1: Multiply equation 1 by 2 and equation 3 by -1.
2(x - y + z) = 2(-1)
-1(2x - y + 2z) = -1(0)
This simplifies the equations to:
2x - 2y + 2z = -2
-2x + y - 2z = 0
Step 2: Add equation 2 and equation 3 to eliminate the x variable.
(x + y + 3z) + (-2x + y - 2z) = -3 + 0
This simplifies to:
-y + z = -3
Step 3: Multiply equation 2 by 2 and add it to equation 1 to eliminate the x variable.
2(x + y + 3z) + (x - y + z) = 2(-3) + (-1)
This simplifies to:
3x + 5z = -7
Step 4: Multiply equation 3 by 2 and add it to equation 1 to eliminate the x variable.
2(2x - y + 2z) + (x - y + z) = 2(0) + (-1)
This simplifies to:
5x + 4z = -2
Step 5: Now we have the simplified system of equations:
-y + z = -3
3x + 5z = -7
5x + 4z = -2
Step 6: Solve equation 1 for y:
y = z - 3
Step 7: Substitute y into equation 2 and equation 3:
3x + 5z = -7
5x + 4z = -2
Step 8: Solve equation 2 for x:
x = (-2 - 4z) / 5
Step 9: Substitute x and y into equation 1:
(-2 - 4z) / 5 - z + 3 = 0
Step 10: Solve equation 9 for z:
(-2 - 4z - 5z + 15) / 5 = 0
(-2 - 9z + 15) / 5 = 0
(-9z + 13) / 5 = 0
-9z + 13 = 0
-9z = -13
z = 13/9
Step 11: Substitute z back into equation 8 to find x:
x = (-2 - 4(13/9)) / 5
x = (-2 - 52/9) / 5
x = (-18/9 - 52/9) / 5
x = (-70/9) / 5
x = -(70/9) * (1/5)
x = -14/9
Step 12: Substitute z and x into equation 6 to find y:
y = z - 3
y = 13/9 - 3
y = 13/9 - 27/9
y = -14/9
Therefore, the solution to the system of equations is:
x = -14/9, y = -14/9, z = 13/9.
what is the solution to the following systems of equations ?
x-y +z=-1
x+y+3z=-3
2x-y+2z=0
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