﻿ MadeFromMultipleAlgebraic1
 CREATED BY: David@TechForText.com ©2010 To contact the author use the email address on the left. The Multiple-Algebraic Calculator The Multiple-Algebraic Calculator demonstrates that a number of complex calculations (involving 24 unknowns, calculus, and trigonometry) can be carried out simultaneously, by devices created with specialized design concepts in the JavaScript or spreadsheet formats. When the user enters 12 numbers, and left clicks with the mouse, the Multiple-Algebraic Calculator initially solves: AX+BY+KZ=M, DX+EY+FZ=N,GX+HY+JZ=W for X, Y and Z. With the calculated values for X, Y, and Z, and the numbers entered by the user, additional sets of calculations are automatically carried out involving six double integrals, ratios, algebra and trigonometry. Instructions To use the Multiple-Algebraic Calculator enter twelve numbers in the twelve white input boxes, below. (To enter or delete numbers, left click with the mouse on the relevant white input box.) For calculated results left click with the mouse on the background Alternatively, with the online JavaScript version you can press the calculation button, and with the spreadsheet version you can press the enter key. Numbers and words displayed in red type are calculated results. Note: If less than 12 numbers are entered this Calculator will NOT function properly. All twelve input boxes must contain numbers entered by the user. To see all the equations, double integrals, trigonometry and calculated results, scroll down. The Multiple-Algebraic Calculator is approximately 20 times the height of a typical computer screen. It is equivalent to a 15 page document. 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. CREATED BY: David@TechForText.com ©2010 To contact the author use the email address on the left.