Solve the system by substitution
-x - y - z = -8
-4x + 4y + 5z = 7
2x + 2z =4
2 x + 2 z = 4 Divide both sides by 2
x + z = 2 Subtract z to both sides
x + z - z = 2 - z
x = 2 - z
Replace x = 2 - z in equation - x - y - z = - 8
- x - y - z = - 8
- ( 2 - z ) - y - z = -8
- 2 + z - y - z = - 8
- 2 - y + z - z = - 8
- 2 - y = - 8 Add 2 to both sides
- 2 - y + 2 = - 8 + 2
- 2 + 2 - y = - 8 + 2
- y = - 6 Multiply both sides by - 1
y = 6
- 4 x + 4 y + 5 z = 7
Replace x = 2 - z and y = 6 in this equation
- 4 * ( 2 - z ) + 4 * 6 + 5 z = 7
- 4 * 2 - 4 * ( - z ) + 24 + 5 z = 7
- 8 + 4 z + 24 + 5 z = 7
- 8 + 24 + 4 z + 5 z = 7
16 + 9 z = 7 Subtract 16 to both sides
16 + 9 z - 16 = 7 - 16
16 - 16 + 9 z = 7 - 16
9 z = - 9 Divide both sides by 9
z = - 1
Replace this value in equation x = 2 - z
x = 2 - z = 2 - ( - 1 ) = 2 + 1 = 3
The solutions are:
x = 3
y = 6
z = - 1
To solve the system by substitution, we need to isolate one variable in one of the equations and substitute it into the other equations. Let's start with the first equation.
1. -x - y - z = -8
Let's isolate x:
-x = -8 + y + z
x = 8 - y - z
Now we have x expressed in terms of y and z.
2. Substitute the expression for x into the other equations:
-4(8 - y - z) + 4y + 5z = 7
2(8 - y - z) + 2z = 4
Expanding and simplifying these equations, we get:
-32 + 4y + 4z + 4y + 5z = 7
16 - 2y - 2z + 2z = 4
Simplifying further, we have:
8y + 9z = 39
-2y = -12
3. Solve for y:
-2y = -12
y = 6
4. Substitute the value of y into one of the equations to solve for z:
8(6) + 9z = 39
48 + 9z = 39
9z = 39 - 48
9z = -9
z = -1
5. Substitute the values of y and z back into one of the original equations to solve for x:
-x - 6 - (-1) = -8
-x - 6 + 1 = -8
-x - 5 = -8
-x = -8 + 5
-x = -3
x = 3
Therefore, the solution to the system is x = 3, y = 6, and z = -1.