Solve the system of equations 3x plus y plus 2z equal 1 also 2x minus y plus z equals negative 3 also x plus y minus 4z equals negative 3?
Given:
3x + y + 2z = 1
2x - y + z = -3
x + y - 4z = -3
Take any one of the equations (we'll use the first one), and
solve for any one of the variables (we'll use y):
y = 1 - 2z - 3x
Now plug that value of y into the latter two equations:
2x - (1 - 2z - 3x) + z = -3
x + (1 - 2z - 3x) - 4z = -3
Now take either of those (again, we'll use the first one), and
solve it for either of the remaining variables (we'll go for
x):
2x - 1 + 2z + 3x + z = -3
∴ 5x = -2 - 3z
∴ x = (3z + 2) / -5
Now take that value, and plug it into our other equation that
uses x and z:
(3z + 2) / -5 + 1 - 2z - 3(3z + 2) / -5 - 4z = -3
Then solve for z:
∴ 2(3z + 2) / 5 - 6z = -4
∴ (6z + 4 - 30z) / 5 = -4
∴ 4 - 24z = -20
∴ 24z = 24
∴ z = 1
Now we can take that value for z, and plug it back into our
previous equation for x and z:
x = (3z + 2) / -5
∴ x = (3 + 2) / -5
∴ x = -1
Finally, we can take those two values, and plug them into our
equation for y:
y = 1 - 2z - 3x
∴ y = 1 - 2 + 3
∴ y = 2
So x = -1, y = 2, and z = 1
You can test these values by plugging them into each of the
original three equations, and seeing if they solve correctly:
3x + y + 2z = 1
∴ 3(-1) + 2 + 2(1) = 1
∴ 2 + 2 - 3 = 1
∴ 1 = 1
2x - y + z = -3
∴ 2(-1) - 2 + 1 = -3
∴ -2 - 2 + 1 = -3
∴ -3 = -3
x + y - 4z = -3
∴ -1 + 2 - 4 = -3
∴ -3 = -3
which shows our answer to be correct.