Error checking and input boxes for AX+BY+KZ = M    
  Accuracy % A= X +  
  Error % B= Y +  
  AX+BY+KZ = K= Z +  
  (AX+BY+KZ) - M = M= =M  
  Z =  
  Above, right is a representation of the equation you entered. If an equation has numbers with more than four  
  digits, look at the vertical representation of the equation, (right side of input boxes, blue numbers and letters)  
   Error checking and input boxes for DX+EY+FZ = N  
  Accuracy % D= X +  
  Error % E= Y +  
  DX+EY+FZ = F= Z +  
  (DX+EY+FZ) - N = N= =N  
  Z =  
  Above, right is a representation of the equation you entered. If an equation has numbers with more than four  
  digits, look at the vertical representation of the equation, (right side of input boxes, blue numbers and letters)  
   Error checking and input boxes for GX+HY+JZ = W  
  Accuracy % G = X +  
  Error % H = Y +  
  GX+HY+JZ = J = Z +  
  (GX+HY+JZ) - W = W = =W  
  Z =  
  Above, right is a representation of the equation you entered. If an equation has numbers with more than four  
  digits, look at the vertical representation of the equation, (right side of input boxes, blue numbers and letters)  
  The Multiple-Algebraic Calculator has a number of error-checking devices (above and below).  
  These devices check the calculations and the numbers you entered to determine if they satisfy  
  the equations. These devices provide feedback in words or numbers. The calculations for  
  all of the error-checking devices are rounded to decimal places. You can change this,  
  by deleting the blue number above, and entering the number of decimal places you prefer. If you enter a number that is too high you may get false indications of errors.  
  Note: The number of zeros displayed is not affected by the blue number.  
     
  For AX+BY+KZ = M, DX+EY+FZ = N, GX+HY+JZ = W  
  The Calculated Results, for X, Y, and Z, are in the three yellow boxes, below  
      X =        
      Y =        
      Z =        
  Spreadsheet formula for X is =(M-(B*Y+K*Z))/A  
  Spreadsheet formula for Y is =(A*N+D*K*Z-A*F*Z-D*M)/(A*E-D*B)  
  Spreadsheet formula for Z is  
  =((A*E-D*B)*D*W-(A*E-D*B)*G*N-((D*H-G*E)*A*N)+(D*H-G*E)*D*M)/((D*H-G*E)*D*K-(D*H-G*E)*A*F-(A*E-D*B)*G*F+(A*E-D*B)*D*J)  
 
  The Multiple-Algebraic Calculator displays a summary of  
  calculated results for 24 unknowns, below. This list does not  
  include all the calculated results. For all the results scroll down.  
  Calculated results on this list are rounded to decimal places. You can change this,  
  by deleting the blue number above, and entering the number of decimal places you prefer.  
  X = Da =  
  Y = Ea=  
  Z = Fa=  
  P = Ga=  
  V = Angle_A=  
  S= Angle_B=  
  T= Ja=  
  Q = Ka=  
  U = La  
  Aa = Ma=  
  Ba = Na=  
  Ca = Pa=  
 
  The three equations from the previous section, (AX+BY+KZ=M, DX+EY+FZ=N,    
  GX+HY+JZ=W are algebraically rearranged by solving for Y as follows:    
      Y= M-AX-KZ   Y= N-DX-FZ   Y= W-GX-JZ        
      B   E   H        
  In this section, calculations with double integrals, will be carried out, for each    
  of above. The calculations are based on the following:    
       
    The X axis will be based 0 to    
    The Z axis will be based 0 to    
                                   
                   
               
                 
        (M-AX-KZ)dxdz =      
        B        
        0      
      0      
  Spreadsheet formula for the above is =(2*M*X*Z-A*Z*(X^2)-K*X*(Z^2))/(2*B)  
                M =  
  Y = M-AX-KZ These are the values A =  
  B you entered in the K =  
        white input boxes. B =  
                     
                 
                   
        (N-DX-FZ)dxdz =      
        E        
        0        
      0      
  Spreadsheet formula for the above is =((2*N*X*Z-D*Z*(X^2)-F*X*(Z^2))/2*E)    
                N=    
  Y= N-DX-FZ These are the values D=    
  E you entered in the F=    
        white input boxes. E=    
                   
               
                 
        (W-GX-JZ)dxdz =    
        H      
        0      
      0      
  Spreadsheet formula for the above is =((2*W*X*Z-D*Z*(X^2)-F*X*(Z^2))/2*H)  
                W=  
  Y= W-GX-JZ These are the values G=  
  H you entered in the J=  
      white input boxes. H=    
  The three equations from the previous section, (AX+BY+KZ=M, DX+EY+FZ=N, and    
  GX+HY+JZ=W have been algebraically rearranged by solving for Z as follows:    
      Z= M-AX-BY   Z= N-DX-EY   Z= W-GX-HY        
      K   F   J        
  In this section, calculations with double integrals, will be carried out, for each    
  of above. The calculations are based on the following:    
       
    The X axis will be based 0 to    
    The Y axis will be based 0 to    
                     
               
                   
        (M-AX-BY)dxdy =      
        K        
        0      
      0      
  Spreadsheet formula for the above is =((2*M*X*Z-A*Z*(X^2)-K*X*(Z^2))/2*B)  
                M =  
  Z = M-AX-BY These are the values A =  
  K you entered in the K =  
        white input boxes. B =  
                     
                 
                   
        (N-DX-EY)dxdy =      
        F        
        0        
      0      
  Spreadsheet formula for the above is =((2*N*X*Z-D*Z*(X^2)-F*X*(Z^2))/2*E)    
                N=    
  Z= N-DX-EY These are the values D=    
  F you entered in the F=    
        white input boxes. E=    
                   
               
                 
        (W-GX-HY)dxdy =    
        J      
        0      
      0      
  Spreadsheet formula for the above is =((2*W*X*Z-D*Z*(X^2)-F*X*(Z^2))/2*H)  
                W=  
  Z= W-GX-HY These are the values G=  
  J you entered in the J=  
        white input boxes. H=    
     
  A Ratio Problem Involving only X as an Unknown  
    In this section: the equations AX+BY+KZ=M, DX+EY+FZ=N, and    
  GX+HY+JZ=W and unknowns X, Y, and Z are converted into a ratio    
    problem, with only one unknown, X. This is possible because the    
    values for X, Y and Z have been calculated.    
  If P*X=Z If V*X=Y    
  then P=Z/X then V=Y/X    
  In terms of numbers In terms of numbers    
  P= V=    
  P*X= V*X=      
  Z= Y=    
    Thus, all of the unknowns represented   Thus, all of the unknowns represented    
    by Z can be replaced by P*X   by Y can be replaced by V*X    
  The three equations (AX+BY+KZ=M, DX+EY+FZ=N, GX+HY+JZ=W)  
  can now be rewritten in terms of X, as follows:  
         
  AX+BY+KZ=M   DX+EY+FZ=N   GX+HY+JZ=W  
  In terms of X   In terms of X   In terms of X  
  AX+BVX+KPX=M   DX+EVX+FPX=N   GX+HVX+JPX=W  
  The above conclusions and calculations can be checked as follows:  
             
        AX+BY+KZ =      
        AX+BVX+KPX =      
        DX+EY+FZ=      
        DX+EVX+FPX=      
        GX+HY+JZ=      
        GX+HVX+JPX=      
           
    AX+BY+KZ=M rewritten in terms of X is AX+BVX+KPX=M      
    This can be restated in the form of a word problem as follows:      
           
    There are three numbers in the ratio of A, B*V, and K*P,        
    and the sum of the three numbers is M.        
               
  In terms of calculated data the three numbers are in the following RATIOS:  
           
      A =    
      B*V =  
      K*P =  
           
    The three numbers are as follows:    
         
      A*X =  
      B*V*X =  
      K*P*X=  
         
    The sum of the numbers is:  
    The above should equal M =    
         
    DX+EY+FZ=N rewritten in terms of X is DX+EVX+FPX=N      
    This can be restated in the form of a word problem as follows:      
           
    There are three numbers in the ratio of D, E*V, and F*P,        
    and the sum of the three numbers is N.      
             
  In terms of calculated data the three numbers are in the following RATIOS:  
         
      D =  
      E*V =  
      F*P=  
         
    The three numbers are as follows:    
         
      D*X =  
      E*V*X =  
      F*P*X=  
         
    The sum of the numbers is:  
    The above should equal N =  
           
    GX+HY+JZ=W rewritten in terms of X is GX+HVX+JPX=W    
    This can be restated in the form of a word problem as follows:    
         
    There are three numbers in the ratio of G, H*V, and J*P,      
    and the sum of the three numbers is W.      
             
  In terms of calculated data the three numbers are in the following RATIOS:  
         
      G =  
      H*V =  
      J*P=  
         
    The three numbers are as follows:    
         
      G*X =  
      H*V*X =  
      J*P*X=  
         
    The sum of the numbers is:  
    The above should equal W =  
                                 
     
  Miscellaneous Calculations: Solving for Additional Unknowns  
     
      This device solves for S        
      A*(E+S)=B*K-Y      
      S =      
      Spreadsheet formula for the above is =(B*K-Y-A*E)/A      
1          
  Spreadsheet formula for above =ROUND(A*(E+S),Rd)=ROUND(B*K-Y,Rd)  
    TRUE = NO Errors for this calculation    
    FALSE = ERROR no result for the numbers you entered.    
     
    This device solves for T    
    X+T+S=A*B*K-Y    
    T=    
    Spreadsheet formula for above =A*B*K-Y-X-S    
       
  Spreadsheet formula for above =ROUND(X+T+S, Rd)=ROUND(A*B*K-Y, Rd)  
  `   TRUE = NO Errors for this calculation    
    FALSE = ERROR no result for the numbers you entered.    
     
    This device solves for U    
    ST-A=B+Z+U    
      U=    
    Spreadsheet formula for above =S*T-A-B-Z    
       
  Spreadsheet formula for above =ROUND(S*T-A,Rd)=ROUND(B+Z+U,Rd)  
    TRUE = NO Errors for this calculation    
    FALSE = ERROR no result for the numbers you entered.    
     
    This device solves for Q    
    T+S-A = (S+Z+U+X)    
    Q    
    Q=    
    Spreadsheet formula for above =(S+Z+U+X)/(T+S-A)    
       
  Spreadsheet formula for above=ROUND(T+S-A, Rd)=ROUND(((S+Z+U+X)/Q),Rd)  
    TRUE = NO Errors for this calculation    
    FALSE = ERROR no result for the numbers you entered.    
         
    This device solves for Aa    
    T*S+X+Y+Z=U+Q+Aa    
    Aa =    
    Spreadsheet formula for above =(T*S+X+Y+Z)-(U+Q)    
       
  Spreadsheet formula for above =ROUND(T*S+X+Y+Z, Rd)=ROUND(U+Q+Aa, Rd)  
    TRUE = NO Errors for this calculation    
    FALSE = ERROR no result for the numbers you entered.    
     
    This device solves for Ba    
    T*S+X+Y+Z=U+Q*Aa*Ba    
    Ba =    
    Spreadsheet formula for above =(T*S+X+Y+Z-U)/(Q*Aa)    
       
  Spreadsheet formula for above=ROUND(T*S+X+Y+Z, Rd)=ROUND(U+Q+Aa*Ba, Rd)  
    TRUE = NO Errors for this calculation    
        FALSE = ERROR no result for the numbers you entered.        
     
    This device solves for Ca    
    Ca*(X+Y+Z)=U+Q+Aa*Ba    
      Ca=      
    Spreadsheet formula for above=(U+Q+Aa*Ba)/(X+Y+Z)    
       
  Spreadsheet formula for above =ROUND(Ca*(X+Y+Z), Rd)=ROUND(U+Q+Aa*Ba, Rd)  
    TRUE = NO Errors for this calculation    
    FALSE = ERROR no result for the numbers you entered.    
     
    This device solves for Da    
    U*Q=U+Q+Aa*Ba+Da    
    Da=    
    Spreadsheet formula for above =U*Q-(U+Q+Aa*Ba)    
       
  Spreadsheet formula for above =ROUND(U*Q,Rd)= ROUND(U+Q+Aa*Ba+Da, Rd)  
    TRUE = NO Errors for this calculation    
    FALSE = ERROR no result for the numbers you entered.    
     
    This device solves for Ea    
    X+Y+Z= Ea*(U+Q+Aa*Ba+Da+Ca)    
    Ea =    
    Spreadsheet formula for above =(X+Y+Z)/(U+Q+Aa*Ba+Da+Ca)    
       
  Spreadsheet formula for above =ROUND(X+Y+Z, Rd)=ROUND(Ea*(U+Q+Aa*Ba+Da+Ca), Rd)  
    TRUE = NO Errors for this calculation    
    FALSE = ERROR no result for the numbers you entered.    
     
    This device solves for Fa    
    A*(Fa+X+Y+Z)= Ea+U+Q+Aa*Ba+Da+Ca    
    Fa =    
    Spreadsheet formula for above=((Ea+U+Q+Aa*Ba+Da+Ca)/A)-(X+Y+Z)    
       
  Spreadsheet formula for above =ROUND(A*(Fa+X+Y+Z), Rd)=ROUND(Ea+U+Q+Aa*Ba+Da+Ca, Rd)  
    TRUE = NO Errors for this calculation    
    FALSE = ERROR no result for the numbers you entered.    
     
    This device solves for Ga    
    A*(Aa+Ba+Ca+Da+Ea+Fa+U+Q) =X+Y+Z+Ga    
    Ga =    
    Spreadsheet formula for above=A*(Aa+Ba+Ca+Da+Ea+Fa+U+Q)-(X+Y+Z)    
       
  Spreadsheet formula for above=ROUND(A*(Aa+Ba+Ca+Da+Ea+Fa+U+Q), Rd)=ROUND(X+Y+Z+Ga, Rd)  
    TRUE = NO Errors for this calculation    
    FALSE = ERROR no result for the numbers you entered.    
     
  Miscellaneous Calculations Solving for trigonometric Unknowns  
     
    This device solves for Angle_A in radians    
    Tan(Angle_A)=Y/X    
    Tan(Angle_A) =    
      Spreadsheet formula for above = Y/X      
    Angle_A=    
    Spreadsheet formula for above = =ATAN(Y/X)    
       
  Spreadsheet formula for above =ROUND(TAN(Angle_A), Rd)=ROUND(Y/X, Rd)  
    TRUE = NO Errors for this calculation    
    FALSE = ERROR no result for the numbers you entered.    
     
  This device solves for Tan(Angle_B), and Angle_A in radians  
    Tan(Angle_B)=X/Y    
    Tan(Angle_B) =    
    Spreadsheet formula for above =X/Y    
    Angle_B=    
    Spreadsheet formula for above =ATAN(X/Y)    
       
  Spreadsheet formula for above =ROUND(TAN(Angle_B), Rd)=ROUND(X/Y, Rd)  
    TRUE = NO Errors for this calculation    
    FALSE = ERROR no result for the numbers you entered.    
     
    This device solves for Ja    
    Ja+Sin(Angle_A)+Cos(Angle_A)=Tan(Angle_B)+A+B    
    Ja =    
  Spreadsheet formula for above=TAN(Angle_B)+A+B-(SIN(Angle_A)+COS(Angle_A))  
       
  Spreadsheet formula for above=ROUND(Ja+SIN(Angle_A)+COS(Angle_A), Rd)=ROUND(TAN(Angle_B)+A+B, Rd)  
    TRUE = NO Errors for this calculation    
    FALSE = ERROR no result for the numbers you entered.    
     
    This device solves for Ka    
    Ka*(Ja+Sin(Angle_A)) = Tan(Angle_B)+A+B+X    
    Ka =    
  Spreadsheet formula for above=(TAN(Angle_B)+A+B+X)/(Ja+SIN(Angle_A))  
       
  Spreadsheet formula for above=ROUND(Ka*(Ja+SIN(Angle_A)), Rd)=ROUND((TAN(Angle_B)+A+B+X), Rd)  
    TRUE = NO Errors for this calculation    
    FALSE = ERROR no result for the numbers you entered.    
     
    This device solves for La    
    La+Cos(Angle_A)=Tan(Angle_B)+Sin(Angle_A)    
    La=    
  Spreadsheet formula for above=TAN(Angle_B)+SIN(Angle_A)-(COS(Angle_A))  
       
  Spreadsheet formula for above=ROUND(La+COS(Angle_A),Rd)=ROUND(TAN(Angle_B)+SIN(Angle_A), Rd)  
    TRUE = NO Errors for this calculation    
    FALSE = ERROR no result for the numbers you entered.    
     
    This device solves for Ma    
    Ma+La*Sin(Angle_A)=Tan(Angle_B)+A+K+D    
    Ma =    
    Spreadsheet formula for above =TAN(Angle_B)+A+K+D-(La*SIN(Angle_A))    
       
  Spreadsheet formula for above=ROUND(Ma+La*SIN(Angle_A),Rd)=ROUND(TAN(Angle_B)+A+K+D, Rd)  
    TRUE = NO Errors for this calculation    
    FALSE = ERROR no result for the numbers you entered.    
     
    This device solves for Na    
    D+Na*Sin(Angle_A) = Tan(Angle_A)+Ma+La+Ma    
    Na =    
  Spreadsheet formula for above=(TAN(Angle_A)+Ma+La+Ma-D)/(SIN(Angle_A))  
       
  Spreadsheet formula for above =ROUND(D+Na*SIN(Angle_A), Rd)=ROUND(TAN(Angle_A)+Ma+La+Ma, Rd)  
    TRUE = NO Errors for this calculation    
    FALSE = ERROR no result for the numbers you entered.    
     
    This device solves for Pa    
    Pa+Cos(Angle_B)=Tan(Angle_B)+Sin(Angle_B)-A-M    
    Pa=    
  Spreadsheet formula for above =TAN(Angle_B)+SIN(Angle_B)-(COS(Angle_B))-A-M  
       
  Spreadsheet formula for above =ROUND(Pa+COS(Angle_B),Rd)=ROUND(TAN(Angle_B)+SIN(Angle_B)-A-M,Rd)  
    TRUE = NO Errors for this calculation    
    FALSE = ERROR no result for the numbers you entered.    
 
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