please help me with the algebra 2 problems and explains how you did it please?

Question

5x + y - z = 6
x + y + z = 2
3x + y = 4

solve for x, y, and z.

Answer 1

The reason you ran awry was because on eof the values was zero ... divide by zero .... (as soon as you had something over z it would go wonky)



y= 6 -5x +z
z= 2 -y -x
x = 2-y -z
x = 2 -1(6 -5x +z) -z
x = 2-6 +5x -2z
4x = 4 +2z ... 8x -8 = 2z

4x -2z = 4 = y +3x
4x -2z = y +3x
x = y -2z = 2 -y -z
... 2y -2 = z
... ... x + y + (2y -2) = 2
... ... x +3y -2 = 2


*****
8x +2y -z = 10

4x +2y +z = 6

4x +2y +z -(5x +y -z) = 0
-x +y +z = 0 ... y+z = x
x + y + z = 2 ... y+z = (2 -x)
2-x = x ... x =1
3x +y = 4
... 3*1 +y = 4 ... y = 0
x +y +z = 2 ... 1 + 0 +z = 2 ... z = 1

Answers:
x = z = 1
y = 0



Sorry about the first part, went in the wrong direction

Answer 2

First, eliminate the z variable in the first two equations. All one has to do is to add the first two equations together to achieve this.

5x + y - z = 6
x + y + z = 2

5x + x is equal to 6x. y + y is equal to 2y. -z + z is equal to zero. 6 + 2 is equal to 8.

So one is left with the equation, "6x + 2y = 8."

Then, one adds this equation together with the 3rd equation.

6x + 2y = 8
3x + y = 4

To do this, convert the the entire 3rd equation into a common coefficient (the number before the variable).

In other words, mulitply the entire "3x + y = 4" by -2. You will get "-6x -2y = -8."

Now, combine this new equation with "6x + 2y = 8."

6x + 2y = 8
-6x - 2y = -8

Of course, this ends up as "0 = 0." Because you have reached this identity, the system of equations has infinitely many solutions.

Uh...I have no idea how to figure out the set of rules that would define exactly what the variable solutions can be. I'm sorry about that; I can't help you here.

Answer 3

You CAN'T solve these because they don't form a basis for 3-space. If you add the first two, you get 6x+2y=8. But this is twice the third equation.

Answer 4

Subtract first two equations.

5x + y - z = 6

x + y + z = 2
_______________

4x - 2z = 4

Notice 3x + y also equals 4

4x - 2z = 3x + y

12x - 6z = 12x + 4y

-6z = 4y

y = -3z/2

3x - 3z/2 = 4

x - z/2 = 2

Judging from here I should assume that there is no solution.

3x - 3z/2 = 6 = 4 ?
__________________

Using catt's explanation along with mine, although I didn't realize it as fast, we have concluded there is no solution. Yay!