The solution to the coordinate system below follows the form (x,y,z). Find the value of the solution. (read the description)

Question

-4x - 5y - z = 18

-2x - 5y - 2z = 12

-2x + 5y + 2z = 4

Answer 1

a) -4x - 5y - z = 18

b) -2x - 5y - 2z = 12

c) -2x + 5y + 2z = 4

Learn the Elimination Method well and you will be ready for matrix operations.

b) -2x - 5y - 2z = 12

c) -2x + 5y + 2z = 4

-------------------------- [Elimination method; add equations]

- 4x = 16

x = -4

a) -4x - 5y - z = 18

b) -2x - 5y - 2z = 12 [Elimination method; subtract equations]

-------------------------

-2x + z = 6

Substitute -4 for x:

-2(-4) + z = 6

z = -2

substitute for x and z in any equation

c) -2x + 5y + 2z = 4

-2(-4) + 5y + 2(-2) = 4

8 + 5y -4 = 4

5y = 0

y = 0

Solution: (-4,0,-2)

CHECK (very important):

Does -4x - 5y - z = 18 ?

-4(-4) -5(0) -(-2) = 18 ?

16 + 2 = 18 ?

18 = 18 ?yes

Does -2x - 5y - 2z = 12 ?

-2(-4) -5(0) -2(-2) = 12 ?

8 + 4 = 12 ?

12 = 12 ?yes

Does -2x + 5y + 2z = 4 ?

-2(-4) + 5(0) + 2(-2) = 4 ?

8 - 4 = 4 ?

4 = 4 ?yes

Answer 2

If you look at the 2nd and 3rd equations closely, notice that if you add them together, the y’s and z’s will be eliminated and you can easily solve for x:

-2x - 5y - 2z = 12

-2x + 5y + 2z = 4

_______________

-4x = 16

x = -4

Now plug that x value into both the 1st and 2nd equations to simplify those to just two variables:

-4x - 5y - z = 18

-2x - 5y - 2z = 12

-4(-4) - 5y - z = 18

16 - 5y - z = 18

-5y - z = 2

-2(-4) - 5y - 2z = 12

8 - 5y - 2z = 12

-5y - 2z = 4

So our two remaining equations are:

-5y - z = 2

-5y - 2z = 4

Since the coefficients in front of the x’s are the same, we can use elimination and subtract the two equations. Just be careful and make sure that you subtract the x’s, the y’s, and the numbers by themselves.

-5y - z = 2

-(-5y - 2z = 4)

____________

z = -2

Finally, plug that z value back into either of the two simplified equations and solve for y:

-5y - z = 2

-5y - (-2) = 2

-5y + 2 = 2

-5y = 0

y = 0

So our solution is x = -4, y = 0, and z = -2 or (-4,0,-2).

Check the solution in all three equations to make sure it works:

-4x - 5y - z = 18

-4(-4) - 5(0) - (-2) = 18

16 - 0 + 2 = 18

18 = 18

-2x - 5y - 2z = 12

-2(-4) - 5(0) - 2(-2) = 12

8 - 0 + 4 = 12

12 = 12

-2x + 5y + 2z = 4

-2(-4) + 5(0) + 2(-2) = 4

8 + 0 - 4 = 4

4 = 4

Answer 3

First, you can get the X coordinate easily by adding together the bottom 2 equations. Notice how eq 2 has -5y and eq 3 has +5y as well as -2z and +2z. When you add them together you are left with -4x = 16, which means that x must be -4.

Now if you plug in x into the top 2 equations, you can easily solve for y and z.

(1) 16 - 5y - z = 18 --> -5y - z = 2

(2) 8 - 5y - 2z = 12 --> -5y - 2z = 4

Now notice how equation 1 and 2 both have -5y terms. That means if we subtract eq 2 from eq 1 we'll be left with a value for z

eq 1 - eq 2 === z = -2

Now that we have x and z, we can plug our values into eq 1 and solve for y.

(1) 16 - 5y + 2 = 18

-5y = 0

y = 0

In conclusion x = -4, y = 0, z = -2

Always remember you can add and subtract equations from one another. Many times this reduces the complexity of the system if you can find terms that cancel out!

Lastly, plug in the values you got to verify your answer!

(1) -4(-4) + 0 + -(-2) = 18

16 + 2 = 18

(2) -2(-4) + 0 + -2(-2) = 12

8 + 4 = 12

(3) -2(-4) + 0 + 2(-2) = 4

8 - 4 = 4