Explanation:

Sure, let's break it down.

- x = 0: This is the y-axis.

- y = 0: This is the x-axis.

- x + y = 1: This line passes through points (1,0) and (0,1).

- 6x + y = 3: This line passes through points (0,3) and (0.5,0).

These lines form a quadrilateral.

- To find the diagonal through the origin:

- The origin is the point (0,0).

- The vertices of the quadrilateral can be found by solving the intersections:

- Intersection of x+y=1 and 6x+y=3 gives a point.

- Intersection of x=0 and x+y=1 is another vertex.

- Intersection of y=0 and 6x+y=3 is another vertex.

- Intersection of x=0 and y=0 is the origin.

The diagonal through the origin should be the line connecting the origin to the intersection of x+y=1 and 6x+y=3:

- Solving x + y = 1 and 6x + y = 3 gives x = 2/5, y = 3/5 as the intersection point.

- The equation of the line (diagonal) passing through (0,0) and (2/5,3/5) needs to be determined.

Let's check each option:

- Option 1: 3x + y = 0

- Option 2: 2x + 3y = 0

- Option 3: 3x - 2y = 0

- Option 4: 3x + 2y = 0

The slope of the desired line is `(3/5)/(2/5) = 3/2`.

Since the slope is 3/2, we can compare this to the options:

- Option 3: 3x - 2y = 0 has the slope of 3/2, which matches.

Thus, the correct answer is:

- Option 3: 3x - 2y = 0

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