Solve the system:

x + y + z = -1
2x + 2y + 5z = 1
5x + 2y + 3z = 8

what is the value of x in the solution?

By what method are you using. If not your calculator on matricies, then Irecommend the method of determinants.

Use simultaneous equation method:

x=-y-z-1...(1)
Now, the equation 2 become:
2(-y-z-1)+2y+5z=1
-2y-2z-2+2y+5z=1
3z=1+2; z=3
Now equation 1 becomes
x=-y-3-1=-y-4; i.e, x=-y-4....(a)
equation 3 becomes:
5(-y-4)+2y+3(3)=8
-5y-20+2y+9=8
-3y=19; y=-19/3
Now the very first equation becomes
x+2(-19/3)+3=1
x=1-3+38/3=38/3-2=12.7-2=10.7
hence x=10.2

To solve this system of equations, we can use the method of elimination or substitution. Let's use the elimination method.

1. Start by multiplying the first equation by 2 and the second equation by 5:
2(x + y + z) = 2(-1) --> 2x + 2y + 2z = -2
5(2x + 2y + 5z) = 5(1) --> 10x + 10y + 25z = 5

2. Subtract the first equation (2x + 2y + 2z = -2) from the second equation (10x + 10y + 25z = 5) to eliminate the x variable:
(10x + 10y + 25z) - (2x + 2y + 2z) = 5 - (-2)
8x + 8y + 23z = 7

3. Multiply the first equation by 5 and the third equation by 2:
5(x + y + z) = 5(-1) --> 5x + 5y + 5z = -5
2(5x + 2y + 3z) = 2(8) --> 10x + 4y + 6z = 16

4. Subtract the first equation (5x + 5y + 5z = -5) from the third equation (10x + 4y + 6z = 16) to eliminate the x variable:
(10x + 4y + 6z) - (5x + 5y + 5z) = 16 - (-5)
5x - y + z = 21

Now we have a new system of equations:
8x + 8y + 23z = 7
5x - y + z = 21

5. Multiply the second equation by 8:
8(5x - y + z) = 8(21) --> 40x - 8y + 8z = 168

6. Add the first equation (8x + 8y + 23z = 7) to the second equation (40x - 8y + 8z = 168) to eliminate the y and z variables:
(8x + 8y + 23z) + (40x - 8y + 8z) = 7 + 168
48x + 31z = 175

Now we have a new equation:
48x + 31z = 175

To find the value of x, we need to isolate x in this equation. Let's continue:

7. Solve for x by subtracting 31z from both sides of the equation:
48x = 175 - 31z

8. Finally, divide both sides by 48 to solve for x:
x = (175 - 31z) / 48

Therefore, the value of x in the solution depends on the value of z in the equation (175 - 31z) / 48.