https://file.aiquickdraw.com/mathbot/file/1726644245214lfr5i5x9.jpg

The contour plot provided shows level curves for a linear function \( L(x, y) \), and the values of the function at specific points are labeled along the curves. In this case, the level curves are straight and parallel, which is characteristic of a linear function of the form:

\[
L(x, y) = ax + by + c
\]

### Step 1: Identify the slope of the level curves
From the contour diagram, observe that the lines are slanted and evenly spaced, suggesting a linear relationship between \( x \) and \( y \). Let’s estimate the slope by choosing two points on one of the lines. For example, from the diagram:

- At \( (0, 6) \), the contour value is \(-75\)
- At \( (6, 0) \), the contour value is \(-75\)

These two points are on the same level curve, indicating the slope between them. The slope is given by the change in \( y \) over the change in \( x \):

\[
\text{slope} = \frac{6 - 0}{0 - 6} = -1
\]

Thus, the slope of the lines is \(-1\).

### Step 2: Construct the equation
The general form of the equation is \( ax + by = \text{constant} \). From the slope, we know that \( a = -b \). Let’s use one of the contour values to find the full equation.

Using the point \( (0, 6) \) where the value is \(-75\):

\[
L(0, 6) = 0 \cdot a + 6 \cdot b = -75
\]

Thus, \( 6b = -75 \), which gives \( b = -12.5 \). Since \( a = -b \), we have \( a = 12.5 \).

### Step 3: Final equation
Substitute the values of \( a \) and \( b \) into the equation:

\[
L(x, y) = 12.5x - 12.5y + c
\]

To find \( c \), use another point, say \( (6, 0) \), where the value is \(-75\):

\[
12.5(6) - 12.5(0) + c = -75
\]

This simplifies to:

\[
75 + c = -75
\]

Thus, \( c = -150 \).

The final equation is:

\[
L(x, y) = 12.5x - 12.5y - 150
\]

Let me know if you would like further details or clarifications! Here are some related questions to explore:

1. How do we determine the spacing between level curves for a linear function?
2. Can a linear function have nonlinear contour plots? Why or why not?
3. What happens to the level curves if the function has a positive constant added or subtracted?
4. How can we confirm that this equation matches the contour plot completely?
5. How would the contour plot change if the slope of the lines were steeper?

**Tip:** For any linear function, the level curves will always be straight lines, and the slope of these lines will give insight into the coefficients of \( x \) and \( y \).

Find a possible equation for a linear function with the given contour diagram.

This problem involves determining the equation of a linear function from a contour plot. Learn how to identify slopes and construct linear equations, a core concept in algebra and calculus.

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