The green line is the line 3y=-x+10

As you can see from the graphs below, there are two possible 
solutions for the rhombus.

   

There is a mistake in your problem, for as you see the line BD cannot
possibly pass through the origin, yet the problem states:

"Show that BD is 3x-y=0"

But 3x-y=0 passes through the origin (0,0), so there is no way that
3x-y=0 could be the equation of BD. 

Let's find what the equation of BD really is.  

BD is parallel to AC and has the same slope.

AC has the equation  

Solving for y,  

thus the slope of AC and BD is 

And BD passes through D(3,9)

Using the point-slope formula:





Multiply through by 3

{{-{x-3)}}} 

{{-x+3}}}



This is correct equation for BD, not 3y-x=0.  

Now we must find the coordinates of the K, the intersection of the
diagonals.  Since the diagonals of a parallellogram bisect each other,
we only need to find the midpoint of one of the diagonals.  We will
need to find the coordinates of C, and then K will be the midpoint of
diagonal CD. 

Since this is a rhombus, all sides must be equal.  So we find side 
AD using the distance formula:

 







So the length of all 4 sides is  

 

Square both sides:


 
Now we must solve this system of equations to find p and q,
the coordinates of C


Solve the first equation for p, p = 10-3q
Substitute in the second equation,







Divide through by 10

Solve by the quadratic formula and get


using the +, substitute in






So point C in the graph on the right above is



using the -, substitute in






So point C in the graph on the left above is



Now we will find the midpoint of CD which will be K,
where the two diagonals intersect.

We will find K for the graph on the left.

For the graph on the left

 and D(3,9)

Using the midpoint formula:

Midpoint = 

Midpoint = 

Midpoint = 

For the graph on the right

 and D(3,9)

Using the midpoint formula:

Midpoint = 

Midpoint = 

Midpoint = 

Edwin