https://file.aiquickdraw.com/mathbot/file/1728718255688ob8zb0v1.jpg

The image provides the following system of equations:

\[
\frac{3}{x} - \frac{4}{y} = -5
\]
\[
2x + y = -5
\]

We are asked to determine the value of \( \text{max}(x) + \text{max}(y) \), where \( \text{max}(x) \) and \( \text{max}(y) \) are the maximum values of \(x\) and \(y\), respectively.

### Step 1: Solve the system of equations

Let's solve this system step by step.

#### Equation 1: 
\[
\frac{3}{x} - \frac{4}{y} = -5
\]
Multiply both sides by \(xy\) to eliminate the denominators:
\[
3y - 4x = -5xy \quad \text{(Equation 1 transformed)}
\]

#### Equation 2:
\[
2x + y = -5
\]

Now, we have the following system:
1. \( 3y - 4x = -5xy \)
2. \( 2x + y = -5 \)

### Step 2: Solve for \( y \) from the second equation

From the second equation:
\[
y = -5 - 2x
\]

### Step 3: Substitute \( y = -5 - 2x \) into the first equation

Substitute this expression for \( y \) into the first equation:
\[
3(-5 - 2x) - 4x = -5x(-5 - 2x)
\]

Simplifying both sides:
\[
-15 - 6x - 4x = -5x(-5 - 2x)
\]
\[
-15 - 10x = -5x( -5 - 2x)
\]
\[
-15 - 10x = 25x + 10x^2
\]

### Step 4: Rearrange the equation to standard form

Move all terms to one side:
\[
10x^2 + 35x + 15 = 0
\]
Divide the whole equation by 5:
\[
2x^2 + 7x + 3 = 0
\]

### Step 5: Solve the quadratic equation

We solve the quadratic equation using the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Here, \(a = 2\), \(b = 7\), and \(c = 3\).

The discriminant is:
\[
\Delta = b^2 - 4ac = 7^2 - 4(2)(3) = 49 - 24 = 25
\]

Now, solving for \(x\):
\[
x = \frac{-7 \pm \sqrt{25}}{2(2)} = \frac{-7 \pm 5}{4}
\]
Thus, we have two solutions for \(x\):
\[
x_1 = \frac{-7 + 5}{4} = \frac{-2}{4} = -\frac{1}{2}, \quad x_2 = \frac{-7 - 5}{4} = \frac{-12}{4} = -3
\]

### Step 6: Solve for \( y \)

Substitute the values of \(x\) back into the equation \(y = -5 - 2x\) to find the corresponding \(y\)-values:

- For \(x = -\frac{1}{2}\):
\[
y = -5 - 2\left(-\frac{1}{2}\right) = -5 + 1 = -4
\]

- For \(x = -3\):
\[
y = -5 - 2(-3) = -5 + 6 = 1
\]

### Step 7: Find the maximum values of \(x\) and \(y\)

The possible solutions are:
\[
(x_1, y_1) = \left(-\frac{1}{2}, -4\right), \quad (x_2, y_2) = (-3, 1)
\]

Thus, the maximum values are:
\[
\text{max}(x) = -\frac{1}{2}, \quad \text{max}(y) = 1
\]

### Step 8: Calculate \( \text{max}(x) + \text{max}(y) \)

\[
\text{max}(x) + \text{max}(y) = -\frac{1}{2} + 1 = \frac{1}{2}
\]

### Final Answer:
The value of \( \text{max}(x) + \text{max}(y) \) is \( \frac{1}{2} \).

---

Do you have any questions or would you like further details?

Here are some related questions to explore:
1. How does the quadratic formula help solve equations in algebra?
2. What are some common methods for solving systems of nonlinear equations?
3. Why do we need to substitute values when solving systems of equations?
4. How can the discriminant help predict the nature of solutions in quadratic equations?
5. How would the solution change if the system had different constants?

**Tip:** Always check the discriminant before using the quadratic formula to know if you’ll have real or complex solutions.

Determine the value of max(x) + max(y) by solving the following system of equations:
(3/x) - (4/y) = -5
2x + y = -5

Explore how to solve a system of equations involving linear and quadratic equations to determine the maximum values of x and y. This detailed solution walks through solving using algebraic techniques like substitution and the quadratic formula. Suitable for students in Grades 9-12.

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