Solve the following system

Question

 

x+y+2z=-2

3x-y+14z=6

x+3y=-5

Please include steps and answer

Answer 1

We will use addition of equations for forward step and back substitution for the solving step.

To make work on the system easier, we denote the first equation R1, the second R2 and third R3:

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

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

x + 3y = -5 --> R3

The equation R3 has coefficient of z equal to 0. We can eliminate the z variable in the second equation by multiplying R1 by 7 and subtract R2:

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

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

7R1 - R2 --> 4x + 8y = -20

The system looks like

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

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

x + 3y = -5 --> R3

(This step makes calculations easier) We divide R2 by 4.

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

The system:

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

x + 2y = -5 --> R2

x + 3y = -5 --> R3

The last step before we start substituting R2 - R3:

R2 --> x + 2y = -5

R3 --> x + 3y = -5

R2 -R3 --> -y = 0.

We see that y = 0.

Now, we substitute y in R2 or R3 and find:

x + 2x0 = -5

x = - 5 - 0 = -5.

Next, we use x and y and equation R1 to find z:

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

2z - 5 = -2,

z = 1.5.

The check of the system is left as an exercise.

I suggest you always check the found solution of a system by substituting all values in the original equations, so you see you did not make a mistake.

Any questions?

Answer 2

You want to create a new system of equations with only two variables. Since the 3rd equation only has two variable, you want to use that one and choose the first two equations and eliminate the z-variable so that it has the same variables at the 3rd equation.

2 goes into 14, but you want -14 + 14 so multiply the first equation by -7 (don't forget BOTH sides):

-7(x + y + 2z) = -7(-2)

3x - y + 14z = 6

Add the two equations together (want the z-variable to cancel):

-7x - 7y - 14z = 14

3x - 1y + 14z = 6

-4x - 8y + 0 = 20

Use the 3rd equation from the original problem and the answer above to create a new system of equations. Solve by either the substitution method or the elimination method:

x + 3y = -5

-4x - 8y = 20

Elmination Method: eliminating the x-variable is easiest --> multiply the first equation by 4

4(x + 3y) = 4(-5)

-4x - 8y = 20

4x + 12y = -20

-4x - 8y = 20

4y = 0

y = 0

Substitute y = 0 and solve for x

x + 3y = -5

x + 3(0) = -5

x = -5

Use either equation 1 or 2 from the original problem, substitute x = -5, y = 0 and solve for z

x + y + 2z = -2

-5 + 0 + 2z = -2

2z = 3

z = 3/2

1. Check the answer by substituting x = -5, y = 0 and z = 3/2 [it must satisfy all three equations].
2. So these three lines intersect at the point (x, y, z) = (-5, 0, 3/2).