solve the following system of equations?

Question

x-3z=-5
2x-y+2z=16
7x-3y-5z=19

Answer 1

#1: x-3z=-5
#2: 2x-y+2z=16
#3: 7x-3y-5z=19

Multiply #2 by (-3):
-6x + 3y - 6z = -48

Now add #2 and #3:
(-6x + 3y - 6z) + (7x - 3y - 5x) = -48 + 19
x - 11z = -29

Now, you have two equations and two unknowns... The one you just found, and equation #1.
x - 11z = -29
x - 3x = -5

Subtract the two equations...
(x - 11z) - (x - 3z) = -29 - (-5)
-8z = -24
z = 3

Now, plug that back into one of the two-variable equations to find x...
x - 3z = -5
x - 3(3) = -5
x - 9 = -5
x = 4

Now, plug x and z into one of the original equations to find y.

2x - y + 2z = 16
2(4) - y + 2(3) = 16
8 - y + 6 = 16
14 - y = 16
-y = 2
y = -2

So, your answers are
x = 4, y = -2, z = 3

Answer 2

if x - 3z = -5 then x = 3z - 5
then,

2(3z - 5) - y + 2z = 16
7(3z - 5) - 3y - 5z = 19

simplify the two equations and add -3 times the first to the second to get z = 3

once you have z you can use either of the two equations you added to get y = -2

with known values of y and z, you can easily find that x = 4 from any of the three original equations

Answer 3

Use Cramer's rule if you know it. Otherwise, do these steps:

1. Solve the first eqn for x (e.g., x = -5 + 3z)
2. Substitute your x answer into eqns 2 and 3 (e.g., 2(-5 + 3z) - y + 2z = 16)
3. You will have eqns 2 and 3 in terms of z and y...two variables, two eqns; so you can use ordinary algebra to solve for z and y.
4. Put your z value into eqn 1 and solve for the x value.

Done. No matrix algebra. You can do the arithmetic.

Answer 4

do u need to find x? or z?

x-3z=-5 x-3z=-5
+3z -x
x=-5 + 3z -3z=-5 - x

2x-y+2z=16
+y
2x+2z=16 + y
either - 2z or - 2x
2x=16+y-2z
0r
2z=16+y-2x

7x-3y-5z=19
+3y
7x-5z=19+3y
either -7x or +5z
7x=19+3y+5z
0r
5z=19+3y-7x

Answer 5

x = -5 + 3z
y = 16 - 2x + 2z

Now, substitute!

-5 + 3z - 48 - 6( - 5 + 3z) - 5z = 19
-5 + 3z - 48 - 30 + 18z - 5z = 19
-83 + 16z = 19
16z = 19 + 83
16z = 102
z = 6.375
x = (6.375 * 3) - 5
x = 14.125
y = 16 - 2x + 2z
y = 16 - (2 * 14.125) + (2 * 6.375)
y= 16 - 28.25 + 12.75
y = 16 - 41
y = -25

So, x = 14.125, y= -25 and z = 6.375.

Answer 6

use a matrix chica.... way easier

Or:
1) Solve first equation for x

2) Plug that expression in for x in the second equation. Solve second equation for y.

3) Plug that expression in for y in the third equation. Also, plug the expression you got for x from equation 1 in the third equation. Third equation should only be a function of z at that point.

4) Solve equation 3 for z.

5) Plug the value of z you found back into equation 2. Solve for y.

6) Plug the z you found in 3 into the first equation. Solve for x.

Answer 7

-6x + 3y - 6z = - 48
7x - 3y - 5z = 19------------ADD

x - 11z = - 29
-x +3z = 5--------------ADD
- 8z = - 24
z = 3
x - 9 = - 5
x = 4
8 - y + 6 = 16
y = - 2

x = 4, y = - 2 , z = 3