I need help solving for all the values in this equation set.

Question

4x + 2y - z = 21

-x - 2y + 2z = 13

3x - 2y + 5z 70

x =

y =

z =

Answer 1

Hi Nnaemeka,

4x + 2y - z = 21

-x - 2y + 2z = 13

3x - 2y + 5z = 70

I would isolate x in the second equation and plug it into the first equation.

-x - 2y + 2z = 13

-x = 13 + 2y - 2z

x = 2z - 2y - 13

4x + 2y - z = 21

4(2z - 2y - 13) + 2y - z = 21

8z - 8y - 52 + 2y - z = 21

7z - 6y = 73

I would then plug that same x into the 3rd equation

3x - 2y + 5z = 70

3(2z - 2y - 13) - 2y + 5z = 70

6z - 6y - 39 - 2y + 5z = 70

11z - 8y = 109

Now take these two equations. Isolate y in the first equation, plug it in for y in the second equation and solve for z.

7z - 6y = 73

11z - 8y = 109

7z - 6y = 73

-6y = 73 - 7z

y = (7/6)z -73/6

11z - 8(7/6 z - 73/6) = 109

11z - 56/6 z + 584/6 = 109

10z/6 = 70/6

10z = 70

z = 7

Now solve for y, and then plug y and z into any of the original equations and solve for x.

7z - 6y = 73

7*7 - 6y = 73

-6y = 24

y = -4

x = 2z - 2y - 13

x = 2(7) - 2(-4) - 13

x = 14 + 8 - 13

x = 22 - 13 = 9

Answer 2

4x + 2y - z = 21 ------------ (1)

-x - 2y + 2z = 13 --------------(2)

3x - 2y + 5z =70 -----------------(3)

Add equations 1 and 2 you will get

3x +0 y + z = 34

Solve for x, x= ( 34 - z ) / 3 ---------(4)

Multiply equation 2 by (-1)

x + 2 y -2z = -13 then add the new equation with equation 3, you will get

4x + 0y + 3z =57

Solve for x, x = ( 57-3z) /4 ----------( 5)

From equation 4 and equation 5 you have:

( 34 - z ) / 3 = ( 57-3z) /4 , using cross product we get:

4(34 - z) = 3 (57 - 3z)

136 - 4z = 171 - 9z , solve for z

5z = 35

z = 7

Plug value of z in equation 5,

x = (57 - 3* 7) / 4

x = 9

Plug values of x and z in equation 1

4*9 + 2y - 7 = 21

36 +2y -7 = 21

29+ 2y = 21

2y = -8

y = -4

Hint ** you have to check your answers in equations 1, 2 & 3 for example,

for equation 1

4*9 + 2 *( -4) - 7 = 36-8-7=36-15=21 so the left side equal the right side.