Rotation of  Cartesian coordinates

<< Calculus`VectorAnalysis`;
<< Graphics`ParametricPlot3D`;
<< Graphics`PlotField`;
<< Graphics`Arrow`;

Remove[a0, d0, c10, line1, dash1]

a0 = {{Thickness[0.003], 
            Arrow[{0, 0}, {8, 0}, 
              HeadCenter -> 0.4 ]
,  
            Arrow[{0, 0}, {0, 7}, 
              HeadCenter -> 0.4   ]}
,  {Thickness[
              0.005], Arrow[{0, 0}, {5, 5}, 
              HeadCenter -> 
                0.4   ]
}
,  {Thickness[0.003], {RGBColor[1, 0, 0], 
              Arrow[{0, 0}, {8, 4}, 
                HeadCenter -> 0.4  ]
, 
              Arrow[{0, 0}, {-3.5, 7}, 
                HeadCenter -> 
                  0.4  ]}}
  };
dline1 = {Dashing[{0.5, 0.0001}], RGBColor[0, 0, 1], 
          Line[{{0, 5}, {5, 5}, {5, 0}}], RGBColor[1, 0, 0], 
          Line[{{5, 5}, {6, 3}}], Line[{{5, 5}, {-1, 2}}], 
          Line[{{5, 0}, {3, 
                4}}]};
d0 = {{RGBColor[0, 0, 0], 
            Disk[{0, 0}, 0.07], 
            Disk[{3, 3}, 0.13]}};
c10 = {{RGBColor[0, 0, 0], Circle[{0, 0}, 1.5, {0, 1.2/8  Pi}], 
              Circle[{5, 0}, 1.3, {Pi/2,  1.30  Pi/2}], 
              Circle[{0, 0}, 
                1.25, {Pi/2, 1.29 Pi/2}]
    }};                
dash1 = {Dashing[{0.02, 0.01}], Line[{{3, 3}, {6, 6}}]};                  
text1={{ RGBColor[0, 0, 0],\(Text[
                StyleForm["Y"
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , 
                  FontSize -> 
                    16
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , 
                  FontWeight -> "Bold"
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , 
                  FontFamily -> "Arial"]
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , \ {0.18, 7.35}]\)
           ,\(Text[
                StyleForm["X"
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , 
                  FontSize -> 
                    
                    16
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , 
                  FontWeight -> "Bold"
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , 
                  FontFamily -> "Arial"]
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , \ {8.3, 0.1}]\)
   ,\(Text[
                StyleForm["O"
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , 
                  FontSize -> 
                    16
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , 
                  FontWeight -> "Plain "
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , 
                  FontFamily -> "Arial"]
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , \ {\(-0.32\), \(-0.01\)}]\)
               ,\(Text[
                StyleForm[\ \*""(Overscript[r, ⇀])""
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , 
                  FontSize -> 
                    18
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , 
                  FontWeight -> "Plain"
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , 
                  FontFamily -> "Times New Roman"]
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , \ {5.31, 5.345}]\)             
,\(Text[
                StyleForm[\ "φ"
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , 
                  FontSize -> 
                    16
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , 
                  FontWeight -> "Plain"
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , 
                  FontFamily -> "Arial"]
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , \ {1.8, 0.41}]\)      
  ,\(Text[
                StyleForm[\ "△ "
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , 
                  FontSize -> 
                    16
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , 
                  FontWeight -> "Plain"
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , 
                  FontFamily -> "Arial"]
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , \ {1.30, 0.60}, {0, 0}, {\(-1\), 1}]\)     
,\(Text[
                StyleForm[\ "φ"
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , 
                  FontSize -> 
                    16
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , 
                  FontWeight -> "Plain"
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , 
                  FontFamily -> "Arial"]
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , \ {4.80, \ 1}]\)     
,\(Text[
                StyleForm[\ "φ"
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , 
                  FontSize -> 
                    16
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , 
                  FontWeight -> "Plain"
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , 
                  FontFamily -> "Arial"]
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , \ {\(-0.35\), 1.41}]\)     
  ,RGBColor[1, 0, 0],\(Text[
                StyleForm[\ "X'"
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , 
                  FontSize -> 
                    16
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , 
                  FontWeight -> "Plain"
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , 
                  FontFamily -> "Arial"]
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , \ {\ 8, 4.5}]\)      
  ,\(Text[
                StyleForm[\ "Y'"
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , 
                  FontSize -> 
                    16
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , 
                  FontWeight -> "Plain"
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , 
                  
                  FontFamily -> "Arial"]
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , \ {\(-3.25\), 7.44}]\)      
  ,Text[StyleForm[ "y cos φ"
                     ,FontSize -> 14
                      ,FontWeight -> "Plain"
                      ,FontFamily -> "Arial"]
                        , {4.23, 2.02},{0, 0},{0.43, -1}]      
,Text[StyleForm[ "x sin φ"
                     ,FontSize -> 14
                      ,FontWeight -> "Plain"
                      ,FontFamily -> "Arial"]
                        , {4.12, 1.12},{0, 0},{0.43, -1}]  
 ,RGBColor[0, 0, 1],\(Text[
                StyleForm[\ "x "
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , 
                  FontSize -> 
                    
                    16
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , 
                  FontWeight -> "Plain"
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , 
                  FontFamily -> "Arial"]
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , \ {5, \(-0.3\)}]\)      
  ,\(Text[
                StyleForm[\ "y"
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , 
                  FontSize -> 
                    14
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , 
                  FontWeight -> "Plain"
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , 
                  FontFamily -> "Arial"]
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , \ {\(-0.323\), 5.1}]\)      
 ,\(Text[
                StyleForm[\ "y"
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , 
                  FontSize -> 
                    14
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , 
                  FontWeight -> "Plain"
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , 
                  FontFamily -> "Arial"]
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , \ {5.2, 2.95}]\)      
 ,Text[StyleForm[ "y sin φ"
                     ,FontSize -> 14
                      ,FontWeight -> "Plain"
                      ,FontFamily -> "Arial"]
                        , {3.81, 4.65},{0, 0},{1, 0.5}]        
 ,Text[StyleForm[ "x cos φ"
                     ,FontSize -> 14
                      ,FontWeight -> "Plain"
                      ,FontFamily -> "Arial"]
                        , {2.1, 1.30},{0, 0},{1, 0.5}]      
 ,Text[StyleForm[ "x cos φ"
                     ,FontSize -> 14
                      ,FontWeight -> "Plain"
                      ,FontFamily -> "Arial"]
                        , {1.131, 3.3180},{0, 0},{1, 0.5}]      
 ,RGBColor[1, 0.0, 0.0],\(Text[
                StyleForm[\ "x'"
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , 
                  FontSize -> 
                    14
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , 
                  FontWeight -> "Plain"
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , 
                  FontFamily -> "Arial"]
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , \ {6, 2.68}]\)     
 ,RGBColor[1, 0, 0],Text[StyleForm[ "∟"
                     ,FontSize -> 30
                      ,FontWeight -> "Plain"
                      ,FontFamily -> "Arial"]
                        , {-0.942, 2.1700},{0, 0},{1. , 0.5}]     
 ,RGBColor[0, 0, 1],Text[StyleForm[ "∟"
                     ,FontSize -> 30
                      ,FontWeight -> "Plain"
                      ,FontFamily -> "Arial"]
                        , {0.1142, 5.1520}]      
 ,RGBColor[0, 0, 1],Text[StyleForm[ "∟"
                     ,FontSize -> 32
                      ,FontWeight -> "Plain"
                      ,FontFamily -> "Arial"]
                        , {0.152, 0.1432}]      
,RGBColor[1, 0, 0],\(Text[
                StyleForm[\ "y'"
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , 
                  FontSize -> 
                    14
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , 
                  FontWeight -> "Plain"
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , 
                  FontFamily -> "Arial"]
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , \ {\(-1.42\), 1.80}]\)     
  ,RGBColor[0.2, 0, 0.8],\(Text[
                StyleForm["Fig.4. Rotation of Cartesian coordinates axes about z-axis

      with angle φ in the direction of counter clock wise\     (ccw)."
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , 
                  FontSize -> 
                    14\ 
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ]
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , \ {\(-3.0\), \(-1.570\)}, {\(-1\), \(-0.5\)}]\)                           }};Show[
        Graphics[Join[a0, dline1, c10, text1]
     , 
          ImageSize -> 550
      , 
          AspectRatio -> Automatic]]  ;

∟

⏜

▲△

(-2 x  + 15) /. 
      x -> 6

Solve[0.5 x  ==  -2 x  + 15]

Solve[{y == 0.5 x  + 2.5, 
        y == -2 x}, {x, y}]

Solve[{y == 0.5 x  + 2.5, 
        y == -2 (x - 5)}, {x, y}]