Need help,adjusting GPS localized coordinates to grid coordinates

This looks like a school problem, so you should have some reference material that helps explain it. Without that background information the question is too vague and open to multiple interpretations.

I am guessing the intent was to do computations using the elevation to move between coordinate systems. However, that's only a guess

A surveyor using RTK could have a data collector set to produce a number of different results. The given numbers look like a State Plane Coordinate system, which is already a grid system. Since you are from NC one would guess the map shows that the problem is set there. The numbers read from the data collector could also be scaled in various ways and if so the numbers should be accompanied by "metadata" to explain how they were scaled. If you also have that information than perhaps the problem can be worked.

I'm a bit scared to answer you... IF I write it out, will I leave out an important detail? There is a BIGGER picture here, and strictly walking you through it, would leave you short. Not to be mean, but can you get someone in person to walk you through it, and show you the bigger picture of Projections, and various coordinate systems?
Are you doing this on a job?
This discussion needs parameters... ie, some basic guidance...

Thank you for your reply.
I also have the drawing to get the grid factor. This is not a job. I am just learning because I didn't work in the States but in Europe and overhere is a bit different. Technical terms and all..

https://www.google.com/search?q=projections+of+state+plane+coordinates&tbm=isch&tbs=rimg:CY43URnvfEghIjiH0a0shbWqZhBGbpBGZnjFzgtMjWv8Fz_1hQ6HrmotPGeOC0wS1wKX_1mghoAbc5JJFcQLRIHdv1CSoSCYfRrSyFtapmEZQeAdP9mmaWKhIJEEZukEZmeMURtpyCsezDipgqEgnOC0yNa_1wXPxGem-G7VTA8-yoSCeFDoeuai08ZEckjHtE6PvgQKhIJ44LTBLXApf8R2mDocSksWIYqEgmaCGgBtzkkkREiJD2hDOwm-SoSCVxAtEgd2_1UJEcLEHhTGBOXj&tbo=u&sa=X&ved=0ahUKEwjzrrKO1IPVAhUEXD4KHVVZBhUQ9C8IHw&biw=1920&bih=1091&dpr=1

Short English lesson; RTK (Real Time Kinematics) not (Real Time Kinetics)

kinematics; the branch of mechanics concerned with the motion of objects without reference to the forces that cause the motion.
kinetics; the study of motion of objects dealing with mass and energy

A better way to describe GPS RTK; (Relative To Known)

Any RTK coordinate is totally useless if the KNOWN point coordinate and coordinate system it is relative to is not given.

Since I cannot see your map, to me your coordinate is useless.

Paul in PA

Howdy Pendula24, welcome
Gave you a couple of likes for your topic and question
Keep trying to upload your maps, it takes a few requirements and several posts to gain that privilege
We would appreciate the opportunity to see

The classic way to handle this is a three step process.
1. Inverse between the reference control point and the located point, which gives you the bearing-distance between the two points.
2. Divide (if the factor is greater than 1) or multiply (if the factor is less than 1) the inversed distance by the given factor which will produce a state plane distance.
3. Calculate the state plane coordinate for the located point by using the referenced control point, the inversed bearing and the state plane distance.

the resulting point will be on-line between the located point and the reference point and slightly closer to the reference point.

The better way is to take the Lat, long for the located point and convert that to state plane, many programs will do this including free ones like corpscon. This should be done even if you went through the three step conversion process to be sure it was done correctly. I wouldn't do the three step calculation I would just import the lat, long numbers into a state plane job to get the coordinates, but you might not have access to those numbers.

The above example ASSUMES that the reference point is actually a point on state plane that the project was expanded around by the given scale factor,,,,this is a huge assumption that needs to be verified. I would not assume that without conformation and I would also check anything I do against the lat, long numbers and the state plane derived from them,,,,at the reference point and located points.

If the factor is less than 1, the ground distance is more than the grid distance. So to get grid distance from ground distance, you multiply by the factor (regardless of whether the factor is less than or greater than 1), and to get ground distance from grid distance, you divide by the factor.

There's an NGS webpage https://www.ngs.noaa.gov/TOOLS/spc.shtml for converting coordinates. For NAD 83, it requires coordinates in meters. Assuming they are in North Carolina (which I am), the feet are US survey feet (1200/3937 m), not international feet (0.3048 m). That gives 163942.5211N 443532.6745E. The lat-lon is thus
35?ø12'50.89233"N 80?ø49'27.36270"W.

Also on the website is https://www.ngs.noaa.gov/NGSDataExplorer/ , which shows the benchmarks in an area. It doesn't have a state plane entry form, but it does have one for lat-lon. Bringing up that lat-lon, I get a map of Charlotte with an X at Hawthorne ?? 5th, a short distance south of Independence Park, where CPCC surveying classes practice. It does not show a benchmark there, but it's close enough to the street to be a likely place for a benchmark that's not on its list. Elevation is 220.779 m, which is consistent with Charlotte (it's 200 m somewhere on the CPCC campus). PK suggests a nail stuck in the ground, rather than a disk with a triangle and dot marked in the center.

Now what you need to do is figure out the local coordinates of the nail, according to the map. I'm not sure about this, because when I draw a map, all the coordinates written on it are grid coordinates, even though all distances are ground distances except one: the tie to the nearest benchmark is dimensioned in both grid and ground. The fixed point (the one whose local coordinates are equal to its grid coordinates) happens to be the other end of the tie from the benchmark, but you can't tell that from looking at the map.