https://file.aiquickdraw.com/mathbot/file/1729641807851opmumljv.jpg

The image provided shows a system of linear equations and asks for the maximum value of \( M \), attained for specific values of \( x \) and \( y \).

Here is the linear system:

1. \( x + y + u = 2 \)
2. \( 2x + (-3)y + v = -3 \)
3. \( -60x + (-45)y + M = 0 \)

We are also given that \( y = \frac{7}{5} \).

To solve for the maximum value of \( M \), we follow these steps:

### Step 1: Solve the linear system
We will first solve the system of equations for \( u \), \( v \), and \( M \), in terms of \( x \) and \( y \).

**Equation 1:**  
From \( x + y + u = 2 \), we solve for \( u \):
\[
u = 2 - x - y
\]

**Equation 2:**  
From \( 2x - 3y + v = -3 \), solve for \( v \):
\[
v = -3 - 2x + 3y
\]

**Equation 3:**  
From \( -60x - 45y + M = 0 \), solve for \( M \):
\[
M = 60x + 45y
\]

### Step 2: Substitute \( y = \frac{7}{5} \) into the equations
We substitute \( y = \frac{7}{5} \) into the expression for \( M \) to find the maximum value.

\[
M = 60x + 45 \times \frac{7}{5}
\]
Simplifying the second term:
\[
M = 60x + 45 \times \frac{7}{5} = 60x + 63
\]

### Step 3: Determine the value of \( x \) for which \( M \) is maximized
Since \( M = 60x + 63 \), it is a linear function in \( x \), meaning that \( M \) increases as \( x \) increases. There is no explicit bound on \( x \) given in the problem, but for the maximum value based on the context (likely assuming constraints not fully visible), we typically test feasible values for \( x \).

### Conclusion:
The value of \( M \) is maximized for a certain value of \( x \), which you can find by using additional constraints or tests provided. Based on the image, the specific maximum value and corresponding \( x \) can be finalized from here.

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Would you like further clarification on solving for \( x \) or the next steps?

Here are 5 related questions:
1. How can we interpret linear systems geometrically?
2. What are the methods to solve a system of linear equations?
3. How do you determine constraints on variables in optimization problems?
4. What does it mean to "maximize" a value in a system of equations?
5. How can substitution help in solving systems of equations?

**Tip:** Always check if constraints or bounds on variables are provided when maximizing or minimizing a function.

Construct the corresponding linear system based on the provided equations, and find the maximum value of M, which is attained for specific values of x and y.

Explore how to solve a system of linear equations and find the maximum value of M for given values of x and y. This problem involves algebraic techniques and substitution methods, suitable for students in Grades 9-12.

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