Solve the following system

Question

x+y+2z=-2

3x-y+14z=6

x+3y=-5

Please include steps and answer

please also try to explain how to draw the system

Answer 1

x = -5 - 3y

Substitute for x in the top two equations.

Eliminate either the y or the z

Determine value of other variable and x.

Answer 2

We will use addition of equations and back substitution to solve the system.

First equation is R1, second equation is R2, third equation is R3.

x + y + 2z = -2 --> R1,

3x - y + 14z = 6 --> R2,

x + 3y = -5 --> R3.

We observe that R3 does not have z, which means coefficient of 0 for z.

We multiply first equation by 7 and subtract the second equation:

7R1 --> 7x + 7y + 14z = -14.

R2 ---> 3x - y + 14z = 6

______________________

4x + 8y = -20.

The system:

x + y + 2z = -2 --> R1,

4x + 8y = -20 --> R2,

x + 3y = -5 --> R3.

We divide R2 by 4. (Note, this makes simply the calculations easier)

R2/4 --> x + 2y = -5.

The system:

x + y + 2z = -2 --> R1,

x + 2y = -5 --> R2,

x + 3y = -5 --> R3.

Now, we simply subtract R3 - R2:

x + 3y = - 5 -->R3

x + 2y = -5 -->R2

y = 0.

We found the variable y. Now, we substitute y in R2 (we can use R3 as well):

x + 2x0 = - 5

We find that x = - 5.

Now, we use R1 to find z:

(-5) + 0 + 2z = -2

and solve for z.

z = 1.5.

You are supposed to check your answers by plugging them in the original equation to verify you did not make a mistake.

As for visualizing solution, it is a point in space.

The three original equations are planes in 3 d coordinate system. They will intersect in a single point with coordinates the solution of the system.

Answer 3