To solve this problem, let's analyze the information provided and the expression we need to evaluate.

Given:

1. $2y + xz = -\frac{2}{3}$
2. $xy = \frac{9}{10}$

We are asked to find the value of $4xy^2 + 2x^2yz$.

Step-by-Step Solution

1. Substitute $xy = \frac{9}{10}$ into the expression: We know that $xy$ is already given, so we’ll try to work with this in the expression we need to evaluate.

2. Express $y$ in terms of $x$ if needed: Since we need $y^2$ and $yz$ terms, let’s explore if we can find values for $y$ and $z$ individually. We currently have:

   • $xy = \frac{9}{10}$
   • $2y + xz = -\frac{2}{3}$

3. Isolate $y$ and $z$: Unfortunately, without additional equations, isolating $y$ and $z$ might not be straightforward. However, if these terms are meant to simplify based on given relationships, we could make a reasonable assumption based on the expression's dependencies.

For clarity and precision, I will continue with detailed algebraic manipulations. Let me calculate these values.From the second equation, $xy = \frac{9}{10}$, we can solve for $y$ as:

$y = \frac{9}{10x}$

We’ll now substitute $y = \frac{9}{10x}$ into the first equation to solve for $z$ in terms of $x$.After substituting $y = \frac{9}{10x}$ into the first equation, we find:

$z = -\frac{2}{30} \cdot \frac{10x + 27}{x^2} = -\frac{10x + 27}{15x^2}$

With expressions for $y$ and $z$ in terms of $x$, we can now substitute these into the target expression $4xy^2 + 2x^2yz$ and simplify.The expression $4xy^2 + 2x^2yz$ simplifies to approximately:

$-1.2$

(The small term $\frac{4.44 \times 10^{-16}}{x}$ is effectively negligible and can be considered zero due to rounding errors.)

Thus, the value of $4xy^2 + 2x^2yz$ is approximately $-1.2$.

Would you like more details on the steps, or do you have any further questions?

Related Questions

1. How would the solution change if $xy$ had a different value?
2. Can you find the individual values of $y$ and $z$ if $x$ is specified?
3. What would happen if we had an additional equation involving $x$, $y$, and $z$?
4. How does the substitution method work in solving systems of equations?
5. Can we use matrix methods to solve this type of problem with more complex equations?

Tip

Always check for any negligible terms that might arise due to floating-point arithmetic when simplifying complex expressions.