Sure, let's add the given algebraic expressions step by step:
First, list the expressions:
1. [tex]\(-4xz - 15xy\)[/tex]
2. [tex]\(20xy - 9yz - 16xz\)[/tex]
3. [tex]\(16xy - 24yz - 16xz\)[/tex]
4. [tex]\(-24xyz\)[/tex]
5. [tex]\(5x^2y^2 - 9yz - 20x^2z^2\)[/tex]
6. [tex]\(5xy - 9yz - 20xz\)[/tex]
### Step 1: Combine like terms
Group all terms with [tex]\(xy\)[/tex]:
[tex]\[
-15xy + 20xy + 16xy + 5xy
\][/tex]
Group all terms with [tex]\(xz\)[/tex]:
[tex]\[
-4xz - 16xz - 16xz - 20xz
\][/tex]
Group all terms with [tex]\(yz\)[/tex]:
[tex]\[
-9yz - 9yz - 24yz - 9yz
\][/tex]
Include the remaining terms as they are:
[tex]\[
-24xyz
\][/tex]
[tex]\[
5x^2y^2
\][/tex]
[tex]\[
-20x^2z^2
\][/tex]
### Step 2: Simplify each group
Simplify the coefficients:
- For [tex]\(xy\)[/tex]:
[tex]\[
-15 + 20 + 16 + 5 = 26 \quad \text{So,} \quad 26xy
\][/tex]
- For [tex]\(xz\)[/tex]:
[tex]\[
-4 - 16 - 16 - 20 = -56 \quad \text{So,} \quad -56xz
\][/tex]
- For [tex]\(yz\)[/tex]:
[tex]\[
-9 - 9 - 24 - 9 = -51 \quad \text{So,} \quad -51yz
\][/tex]
### Step 3: Combine all simplified terms
After combining all terms, the resulting expression is:
[tex]\[
5x^2y^2 - 20x^2z^2 - 24xyz + 26xy - 56xz - 51yz
\][/tex]
Therefore, the simplified expression obtained by adding all the given algebraic expressions is:
[tex]\[
5x^2y^2 - 20x^2z^2 - 24xyz + 26xy - 56xz - 51yz
\][/tex]
