If 4 x + 10xy + 10y =45 then find the minimum value of 2x+5y​

To find the minimum value of \(2x + 5y\), given the equation \(4x + 10xy + 10y = 45\), we can approach this using the method of Lagrange multipliers or by substitution. However, given the linear and quadratic nature of the equation, let's simplify by substituting \(z = 2x + 5y\) and then solve for the minimum value.

### Step 1: Express the equation in terms of \( z \)
Let's substitute \( x = \frac{z - 5y}{2} \) into the equation \(4x + 10xy + 10y = 45\).

### Step 2: Substitute and simplify
Substituting \( x = \frac{z - 5y}{2} \) into the equation:
\[
4\left(\frac{z - 5y}{2}\right) + 10\left(\frac{z - 5y}{2}\right)y + 10y = 45
\]

Simplify each term:
\[
2(z - 5y) + 5(z - 5y)y + 10y = 45
\]
\[
2z - 10y + 5zy - 25y^2 + 10y = 45
\]
Combine like terms:
\[
2z + 5zy - 25y^2 = 45
\]

### Step 3: Solve for z
Now, we solve for \( z \) to find its minimum value. This expression can be quite complex, so another way to find the minimum value is to take partial derivatives with respect to \( x \) and \( y \), and set them to zero.

### Step 4: Use partial derivatives and find critical points
To minimize \( z = 2x + 5y \), we differentiate the equation with respect to \( x \) and \( y \):

\[
\frac{\partial z}{\partial x} = 2 + 10y = 0
\]

\[
\frac{\partial z}{\partial y} = 10x + 10 = 0
\]

From \( \frac{\partial z}{\partial y} = 0 \), \( 10x + 10 = 0 \Rightarrow x = -1 \).

Substituting \( x = -1 \) into the original equation \( 4x + 10xy + 10y = 45 \):
\[
4(-1) + 10(-1)y + 10y = 45
\]
\[
-4 - 10y + 10y = 45 \quad \text{which simplifies to a contradiction.}
\]

Therefore, to find the minimum value directly, consider testing critical values or using optimization techniques numerically, or analyze further through substitutions or other numerical methods.

### Conclusion:
The minimum value \( z = 2x + 5y \) requires further algebraic simplification or a specific method like Lagrange multipliers or numerical methods.

Would you like to delve into numerical methods, or should I continue with algebraic simplification? Alternatively, do you have any specific parts you need further clarification on?

### Relative Questions:
1. How can we apply Lagrange multipliers to find minimum values in constrained optimization problems?
2. What are the advantages of using substitution methods in solving simultaneous equations?
3. Could there be an easier approach to solve quadratic forms like this?
4. How do partial derivatives help in optimization problems?
5. What specific numerical methods can be applied for minimizing complex equations?

### Tip:
When dealing with complex optimization problems, breaking down the equation into smaller parts or applying a known method, like substitution or Lagrange multipliers, can significantly simplify the problem.

Learn how to find the minimum value of 2x + 5y given the equation 4x + 10xy + 10y = 45. This problem involves optimization techniques and the use of Lagrange multipliers. Suitable for advanced high school students and beyond.

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