simulanious equations Help?

Let's do the first problem.

From the first equation: 2x + y = 8, so y = -2x + 8
Plugging this into the second equation:
(-2x + 8) + 3z = 5
This simplifies to
-2x + 3z = -3

From the third equation: z+w = 1, so w = -z + 1
Plugging this into the fourth equation:
5*(-z + 1) + 3x = 9
This simplifies to
3x - 5z = 4

Now you have two equations in terms of x and z:
-2x + 3z = -3
3x - 5z = 4

Now let's solve this system of equations. First, let's multipy the equations to get convenient coefficients for x:
-6x + 9z = -9
6x -10z = 8
Adding these two equations together, we get:
-z = -1
so z = 1

And hence, x = 3
And hence, y = 2
And w = 0

Hopefully that helps!

I'll let you do the second one yourself. The trick is just to combine equations, two at a time, to eliminate a variable.

To solve this you can use substitution or if you know about matrices you can generate an augmented matrix and do row reduce echelon form (rref).
For the first set of equations the answers are:
x=3, y=2, z=1, w=0

For second set of equations the answers are:
x=4, y=3, z=2, w=1

I) 2x+y=8 ==> y = 8-2x
II) y+3z=5 ==> y = 5 - 3z
III) z+w=1 ==> z = 1 - w and w = 1 - z
IV) 5w+3x=9 ==> 3x = 9 - 5w; x = (9-5w)/3
-------
I and II)
(8 - 2x) = (5 - 3z)
-2x + 3z = 5 -8
-2x + 3z = -3 (-)
2x - 3z = 3
---
2x - 3(1-w) = 3
2x - 3 + 3w = 3
2x +3w = 3 + 3
2x + 3w = 6
----
2x + 3(1-z) = 6
2x + 3 - 3z = 6
2x -3z = 6 - 3
2x - 3z = 3
---
2x - 2z = 3
2(9-5w)/3 - 3z = 3
18 - 10w - 9z = 9
-10w -9z = 9-18
-10w - 9(1-w) = -9
-10w -9 + 9w = -9
-w = -9 + 9
w = 0

z = 1- w
z = 1 - 0
z = 1

y = 5 - 3(1)
y = 5 - 3
y = 2

y = 8 - 2x
2 = 8 - 2x
2 - 8 = -2x
-6 = -2x
x = 6/2
x = 3
Ans: x = 3, y = 2, z = 1, w = 0
<>
Answers for these one too.
y+z+w=6
z+2w+x=8
3w+x+y=10
x+y+z=9
------
eliminating:
y + z + w = 6 (I)
-y -z - x = -9 (II)
-----------------
w -x = -3
w = -3 + x
---
3w + x + y = 10 (III)
w - x = 3
----------------------
4w + y = 13
y = 13 - 4w
------
z + 2w + x = 8 (II)
z + 2(-3+x) +x = 8
z -6 + 2x + x = 8
z + 3x = 8 + 6
z + 3x = 14
z = 14 - 3x
---
x+y+z = 9
x + (13-4w) + (14-3x) = 9
x + 13 - 4w + 14 - 3x = 9
-2x + 27 -4w = 9
-2x -4(-3 +x) = 9 - 27
-2x + 12 -4x = -18
-6x = 18-12
x = 30/6
x = 5
---
z = 14 - 3(5)
z = 14 - 15
z = -1
---
w = -3 + (5)
w = 2
---
y = 13 - 4(2)
y = 13 - 8
y = 5
Answers:
x = y = 5; w = 2; z = -1
<>`´<>

First Question:
Steps:
1. Make all the left side of the equation has one variable left, i.e.: y=8-2x, y=5-3z, w=1-z.
2. Then equate y=8-2x and y=5-3z
3. You will make z= whatever is on the right hand side
4. Finally, replace z in the last equation and you will get the answer at the end.

I don't quite understand 3 equations with y gone, do you mean to start it by;

2x+y=8 > y=8-2x
y+3z=5 > y=5-3z
so,
8-2x=5-3z