Alright! Let's go through the step-by-step process to simplify the given expression \(\left(3 x y^3 + 8 x^2 y - 3 x y\right) + \left(7 x y^3 - 9 x^2 y + 6 x y\right)\).
### Step 1: Write down the expression
The expression to simplify is:
[tex]\[ (3 x y^3 + 8 x^2 y - 3 x y) + (7 x y^3 - 9 x^2 y + 6 x y) \][/tex]
### Step 2: Combine like terms
To simplify the expression, we need to combine the like terms. Like terms are terms that contain the same variables raised to the same power. Here we have three types of terms: \(x y^3\), \(x^2 y\), and \(x y\).
#### Combine the \(x y^3\) terms:
[tex]\[ 3 x y^3 + 7 x y^3 = (3 + 7) x y^3 = 10 x y^3 \][/tex]
#### Combine the \(x^2 y\) terms:
[tex]\[ 8 x^2 y - 9 x^2 y = (8 - 9) x^2 y = -1 x^2 y \][/tex]
#### Combine the \(x y\) terms:
[tex]\[ -3 x y + 6 x y = (-3 + 6) x y = 3 x y \][/tex]
### Step 3: Put all the like terms together
After combining the like terms, we get:
[tex]\[ 10 x y^3 - 1 x^2 y + 3 x y \][/tex]
### Step 4: Write the final simplified expression
Putting it all together, the simplified expression is:
[tex]\[ 10 x y^3 - x^2 y + 3 x y \][/tex]
This is the simplified form of the given expression.
