Problem on relative motion involving wind speed .Finding the real velocity of wind?

Question

Here is the question

For a person running west at 7km/hr wind appears to blow from north-west .But when he walks towards west at 3km/hr the wind appears to blow from the north. What is the true velocity and direction of the wind??

I have tried to solve it and I 've gone through this way
First I derived two equations

$V_{W/M}=V_{W/E}-V_{M/E}$
$V_{W/M}+V_{M/E}=V_{W/E}$

Therefore the velocity triangle for the first instance will be

For the second instance

here i've got stuck and i dont know how to solve it .There are three unknowns and with 2 velocity triangle . Please refer to my steps and help me to solve it

Note: The velocity triangle in the first instance should be marked a 45deg in between the two vectors because of the north-west direction.

Answer 1

You don't need to go into any serious algebra to solve this problem, just think about the vectors and how they manipulate under changes in your reference frames.

Reference Frames

Let's say you and your twin are standing at a train station and we see two (VERY LONG) trains coming into the station on parallel tracks at the same time from opposite directions at 5 km/hr, and then they suddenly stop. You saw one of their speeds being 5 km/hr in the WEST direction (lets call that +x), and the other being 5 km/hr in the EAST direction (lets call that the -x direction). We can also say that the second train was travelling -5 km/hr in the WEST direction.

Now let's only you get on the first train, and both trains abruptly continue down the track going 5 km/hr in their respective directions. You have entered a new reference frame, where you are now traveling 5 km/hr in the WEST direction relative to your twin's frame (the rest frame). This also means that your twin looks like he's going 5 km/hr in the EAST direction relative to you.

IF your twin had gotten on the other train and you were looking at him, he would appear to being moving 10 km/hr to the EAST relative to your frame. You would look like you are moving 10 km/hr to the WEST in his reference frame.

The way you can mathematically represent this changing frame is by adding a vector to all your objects that is equal to the change in velocity between your reference frames.

Conventions

For the purposes of this question the x direction is the direction you are moving, y-direction upwards. The frames I'm using are the 7 3 and rest frames, 7 means that if you are running at 7 km/hr parallel to your vector it would stay the same distance away from you the whole time.

The Problem

In the 3 frame there is no x component of the wind, it is going at the same relative speed as you (3 km/hr in the x-direction when compared against a rest frame)

Let's go from the 3 frame to the 7 frame to solve for the y component of the wind now.

When you go 4 km/hr faster in the the x-direction the wind in the x-direction appears to go -4 km/hr. This makes a 45-45-90 triangle, telling you that your y (which doesn't change in any of the 3 frames, is equal in magnitude to -4 km/hr.

So your wind ends up being $<3,-4>$ in the rest frame