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?

Answer 1

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.