Question

HELP WITH SOLVING SYSTEMS WITH THREE VARIABLES, WILL MEDAL

Answer 1

Equation 1)  -x - y - z = -8
Equation 2)  -4x + 4y + 5z = 7
Equation 3)  2x + 2z = 4

Solving for three variables is quite similar to solving for two variables. Just like when solving for two variables, you look for like variables & solve them one-by-one. So, let's get started! :)

Multiply ALL of equation 1 by four.

1)  4(-x - y - z = -8)

Simplify.

1)  -4x - 4y - 4z = -32

Both equations 1 & 2 have the same three variables (x, y, z), so, we must look at these equations together.

1)  -4x - 4y - 4z = -32
2)  -4x + 4y + 5z = 7

Add these equations together, and create a fourth equation.

4)  -8x - z = -25

Now, notice how both equations 3 & 4 have the same two variables (x, z).

So, we must look at these equations together.

3)  2x + 2z = 4
4)  -8x - z = -25

Multiply ALL of equation 4 by 2.

4)  2(-8x - z = -25)

Simplify.

4)  -16x - 2z =-50
3)  2x + 2z = 4

Now, add the equations together, so that we can solve for x.

-14x = -42

Divide both sides by -14.

x = 3

Now, plug in 3 for x into our 3rd equation.

3)  2x + 2z = 4

3)  2(3) + 2z = 4

Simplify.

6 + 2z = 4

Subtract 6 from both sides.

2z = 4 - 6

Simplify.

2z = -2

Divide both sides by 2.

z = -1

Now, plug in -1 for z & 3 for x in our 1st equation.

1)  -x - y - z = -8

-(3) - y - (-1) = -8

Simplify.

-3 - y + 1 = -8

Simplify.

-2 - y = -8

Add 2 to both sides.

-y = -8 + 2

Simplify.

-y = -6

Divide both sides by -1.

y = 6

SO :

x = 3, y = 6, z = -1

~Hope I helped!~