Sure! Let's solve the given expression step-by-step.
We have the expression:
[tex]\[ 8x - (15y + 16z - 12x) - (-13x + 20y) - (x + y + z) \][/tex]
### Step 1: Distribute the negative signs inside the parentheses
First, let's remove the parentheses by distributing the negative signs where required:
[tex]\[ 8x - (15y + 16z - 12x) = 8x - 15y - 16z + 12x \][/tex]
[tex]\[ - (-13x + 20y) = + 13x - 20y \][/tex]
[tex]\[ - (x + y + z) = -x - y - z \][/tex]
So, combining all of these, we get:
[tex]\[ 8x - 15y - 16z + 12x + 13x - 20y - x - y - z \][/tex]
### Step 2: Combine like terms
Next, let's group and combine the like terms:
#### Combine the [tex]\(x\)[/tex] terms:
[tex]\[ 8x + 12x + 13x - x \][/tex]
[tex]\[ = (8 + 12 + 13 - 1)x \][/tex]
[tex]\[ = 32x \][/tex]
#### Combine the [tex]\(y\)[/tex] terms:
[tex]\[ -15y - 20y - y \][/tex]
[tex]\[ = (-15 - 20 - 1)y \][/tex]
[tex]\[ = -36y \][/tex]
#### Combine the [tex]\(z\)[/tex] terms:
[tex]\[ -16z - z \][/tex]
[tex]\[ = (-16 - 1)z \][/tex]
[tex]\[ = -17z \][/tex]
### Conclusion
After combining all the terms, the final, simplified expression is:
[tex]\[ 32x - 36y - 17z \][/tex]
So the solution to the given expression is:
[tex]\[ 32x - 36y - 17z \][/tex]
