document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=67306>Question 67306</A>This question is from textbook <A HREF=http://www.amazon.com/exec/obidos/ASIN/093979862x/algebrahomeworkh>An Incremenatl Development</A><BR>\n" );
document.write( ": <I><SPAN STYLE=\"word-break: break-all;\"> Please help to solve :\r<BR>\n" );
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document.write( "2x + 3y - 2z = -3<BR>\n" );
document.write( "x + 9y - 2z = 3<BR>\n" );
document.write( "3x + 6y + 2z = -1 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #47897 by </B><A HREF=/tutors/aboutme.mpl?userid=venugopalramana><B>venugopalramana(3286)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=venugopalramana><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.geocities.com/venugopalramana/FREE_MATHS_ASSISTANCE.html valign=middle><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=47897\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> SEE THE FOLLOWING EXAMPLE AND TRY. IF IN DOUBT PLEASE COME BACK\r<BR>\n" );
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document.write( "I need help with usings Cramer's Rule. The question<BR>\n" );
document.write( "is: Solve the following system of equations using<BR>\n" );
document.write( "Cramer's Rule.<BR>\n" );
document.write( "2x + 4y - 3z = 2<BR>\n" );
document.write( "2x - 2y + 3z = 3<BR>\n" );
document.write( "3x - 4y + 5z = 4<BR>\n" );
document.write( "I understand that I have to first have to set up the<BR>\n" );
document.write( "determinants. Which are<BR>\n" );
document.write( "D = 2 4 -3 Answer Column is 2<BR>\n" );
document.write( "2 -2 3 3<BR>\n" );
document.write( "3 -4 5 4<BR>\n" );
document.write( "------------------------------------------------------------------<BR>\n" );
document.write( "OK…BUT LET US INTRODUCE SOME NOMENCLATURE TO MAKE IT<BR>\n" );
document.write( "EASY TO REMEMBER.<BR>\n" );
document.write( "FIRST MAKE D= DETERMINANT OF COEFFICIENTS.THAT IS<BR>\n" );
document.write( "WRITE COEFFICIENTS OF X,Y AND Z IN THE SAME ORDER IN<BR>\n" );
document.write( "ALL 3 EQNS.AS SHOWN BELOW. WE PUT THEM APART TO WRITE<BR>\n" );
document.write( "COLUMN NUMBERS.<BR>\n" );
document.write( "COLUMN I(X)....COL.II(Y)...COL.III(Z)<BR>\n" );
document.write( "2,................4,.........-3..............I ROW<BR>\n" );
document.write( "2,...............-2,..........3...............II ROW<BR>\n" );
document.write( "3,...............-4,..........5..................III<BR>\n" );
document.write( "ROW<BR>\n" );
document.write( "OK ...NOW WE SHALL FIND ITS VALUE...WE DO IT IN A<BR>\n" );
document.write( "STEP-WISE PROCESS AS GIVEN BELOW..THERE ARE 3 ROWS AND<BR>\n" );
document.write( "3 COLUMNS.WE CAN FIND ITS VALUE USING ANY ONE ROW OR<BR>\n" );
document.write( "COLUMN.LET US USE FIRST COLUMN TO EXPAND IN THIS<BR>\n" );
document.write( "EXAMPLE.FIRST COLUMN HAS 3 ELEMENTS...2,2,AND 3.LET US<BR>\n" );
document.write( "CALL THEM E11,E21 AND E31 TO SHOW THAT THEY ARE FROM<BR>\n" );
document.write( "ROW1,COL.1;ROW2,COL.1;AND ROW3,COL.1.THE FORMULA FOR<BR>\n" );
document.write( "VALUE OF DETERMINANT IS<BR>\n" );
document.write( "D=(E11)*(S11)*(M11)+(E21)*(S21)*(M21)+(E31)*(S31)*(M31)……………………………………..I<BR>\n" );
document.write( "THAT IS WE MULTIPLY EACH ELEMENT WITH ITS CO-FACTOR<BR>\n" );
document.write( "AND ADD ALL THE RESULTS.COFACTOR OF AN ELEMENT IS<BR>\n" );
document.write( "PRODUCT OF 2 NUMBERS...ONE IS FOR SIGN(S11,S21,S31)<BR>\n" );
document.write( "AND ANOTHER FOR VALUE(M11,M21,M31).<BR>\n" );
document.write( "THE NUMBER FOR SIGN IS (-1)^(ROW NUMBER+COLUMN<BR>\n" );
document.write( "NUMBER)OF THE ELEMENT.HERE THE FIRST ELEMENT 2 IS FROM<BR>\n" );
document.write( "ROW1 AND COL.1.SO ITS SIGN NUMBER IS<BR>\n" );
document.write( "(-1)^(1+1)=(-1)^2=+1=S11<BR>\n" );
document.write( "THE NUMBER FOR VALUE IS CALLED THE MINOR OF THE<BR>\n" );
document.write( "ELEMENT.IT IS OBTAINED BY DELETING THE ROW AND COLUMN<BR>\n" );
document.write( "CONTAINING THE ELEMENT.OUR ELEMENT IS 2 FROM ROW1 AND<BR>\n" );
document.write( "COLUMN 1, AS GIVEN ABOVE.SO REMOVE THE ROW1 AND COLUMN<BR>\n" );
document.write( "1.SO WE GET A 2 BY 2 DETERMINANT.IT IS<BR>\n" );
document.write( "-2,3<BR>\n" );
document.write( "-4,5<BR>\n" );
document.write( "LET US CALL THIS EQUAL TO M11 TO INDICATE IT IS MINOR<BR>\n" );
document.write( "OF ELEMENT 11 (E11)THAT IS ROW 1 AND COLUMN 1.<BR>\n" );
document.write( "SO THE FIRST ELEMENT GIVES US ITS CONTRIBUTION<BR>\n" );
document.write( "AS...(2)*(+1)(M11)...WE SHALL FIND THE VALUE OF M11 IN<BR>\n" );
document.write( "THE SAME WAY AS ABOVE.BUT LET US SKIP THIS FOR A<BR>\n" );
document.write( "MOMENT AND COMPLETE OUR JOB.<BR>\n" );
document.write( "NOW TAKING THE NEXT NUMBER 2 AGAIN WHICH IS FROM ROW 2<BR>\n" );
document.write( "AND COL.1.SO ITS<BR>\n" );
document.write( "SIGN =S21=(-1)^(2+1)=(-1)^3=-1<BR>\n" );
document.write( "MINOR IS = M21 GIVEN BY<BR>\n" );
document.write( "4,-3<BR>\n" );
document.write( "-4,5<BR>\n" );
document.write( "SO CONTRIBUTION OF E21 IS<BR>\n" );
document.write( "(2)(-1)(M21)<BR>\n" );
document.write( "SIMILARLY CONTRIBUTION OF E31 IS<BR>\n" );
document.write( "(3)*{(-1)^(3+1)}*(M31)=(3)*(1)*(M31) WHERE M31 IS THE<BR>\n" );
document.write( "2 BY 2 DETERMINANT<BR>\n" );
document.write( "4,-3<BR>\n" );
document.write( "-2,3<BR>\n" );
document.write( "SO NOW WE GOT THE 3 BY 3 DETERMINANT CONVERTED TO 2 BY<BR>\n" );
document.write( "2 DETERMINANT.THE SAME METHOD WE CAN USE TO EVALUATE<BR>\n" );
document.write( "ANY ORDER DETERMINANT BY REDUCING IT TO A LOWER ORDER<BR>\n" );
document.write( "DETERMINANT IN EACH STEP.4 BY 4 TO 3 BY 3 TO 2 BY 2<BR>\n" );
document.write( "WHICH IS THE END AS WE SHOW BELOW.<BR>\n" );
document.write( "LET US FINISH THIS NOW BY ONE EXAMPLE OF 2 BY 2<BR>\n" );
document.write( "DETERMINANT SAY M11<BR>\n" );
document.write( "-2,3<BR>\n" );
document.write( "-4,5<BR>\n" );
document.write( "M11=BY THE SAME METHOD<BR>\n" );
document.write( "….=(-2)(1)(5)+(-4)(-1)(3)=-10+12=2<BR>\n" );
document.write( "SO BY THIS WAY WE CAN FIND M21 AND M31 AND HENCE FIND<BR>\n" );
document.write( "D BY THE ABOVE FORMULA-I...VIZ..<BR>\n" );
document.write( "D=(E11)*(S11)*(M11)+(E21)*(S21)*(M21)+(E31)*(S31)*(M31)……………………………………..I<BR>\n" );
document.write( "NEXT MAKE DX…..BY REPLACING COEFFICIENTS OF X IN D<BR>\n" );
document.write( "WITH CONSTANT TERMS…SO DX IS<BR>\n" );
document.write( "COLUMN I(CONSTANTS)....COL.II(Y)...COL.III(Z)<BR>\n" );
document.write( "2,.………………………...............4,.........-3..............I<BR>\n" );
document.write( "ROW<BR>\n" );
document.write( "3,………………………...............-2,..........3...............II<BR>\n" );
document.write( "ROW<BR>\n" );
document.write( "4,...………………………............-4,..........5..................III<BR>\n" );
document.write( "ROW<BR>\n" );
document.write( "-------------------------<BR>\n" );
document.write( "D of x = 2 4 -3<BR>\n" );
document.write( "3 -2 3<BR>\n" );
document.write( "4 -4 5<BR>\n" );
document.write( "SO YOURS IS OK TOO<BR>\n" );
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document.write( "SIMILARLY DY IS<BR>\n" );
document.write( "COLUMN I(X)....COL.II(CONSTANT)...COL.III(Z)<BR>\n" );
document.write( "2,…………………….............…2,.........-3..............I<BR>\n" );
document.write( "ROW<BR>\n" );
document.write( "2,...…………………….........…3,..........3...............II<BR>\n" );
document.write( "ROW<BR>\n" );
document.write( "3,........…………………….......4,..........5..................III<BR>\n" );
document.write( "ROW --------------------------------<BR>\n" );
document.write( "D of y = 2 2 -3<BR>\n" );
document.write( "2 3 3<BR>\n" );
document.write( "3 4 5<BR>\n" );
document.write( "SO YOURS IS OK<BR>\n" );
document.write( "----------------------<BR>\n" );
document.write( "SIMILARLY DZ IS<BR>\n" );
document.write( "COLUMN I(X)....COL.II(Y)...COL.III(CONSTANT)<BR>\n" );
document.write( "2,...........………….....4,......….2..............I ROW<BR>\n" );
document.write( "2,.........…………......-2,..........3...............II<BR>\n" );
document.write( "ROW<BR>\n" );
document.write( "3,......……………......-4,.......…4..................III<BR>\n" );
document.write( "ROW<BR>\n" );
document.write( "-----------------------<BR>\n" );
document.write( "D of Z = 2 4 2<BR>\n" );
document.write( "2 -2 3<BR>\n" );
document.write( "3 -4 4<BR>\n" );
document.write( "SO YOURS IS OK TOO<BR>\n" );
document.write( "------------------------------<BR>\n" );
document.write( "Now my problem is how do I evaluate the determinants?<BR>\n" );
document.write( "I think that I understand how to set it up. I'm just<BR>\n" );
document.write( "not sure what to do after that. Any help you can give<BR>\n" );
document.write( "me would be greatly appreciated. I'm not necessarily<BR>\n" );
document.write( "looking for the entire question, just an explanation<BR>\n" );
document.write( "of what to do next. I've looked over the web for hours<BR>\n" );
document.write( "and none of the Algebra websites explains how to do<BR>\n" );
document.write( "this step. Thanks for the help and have a great night.<BR>\n" );
document.write( "Jonna<BR>\n" );
document.write( "HOPE NOW YOUR PROBLEM IS SOLVED AS FAR AS EVALUATING<BR>\n" );
document.write( "THE DETERMINANTS IS COCERNED.LET US FINISH BY GIVING<BR>\n" );
document.write( "FINAL FORMULAE TO FIND X,Y AND Z<BR>\n" );
document.write( "X=DX/D……..;…..Y=DY/D……………..;<BR>\n" );
document.write( "AND………………….Z=DZ/D………………………………….II<BR>\n" );
document.write( "----------------------------------------------------------------------------------<BR>\n" );
document.write( "NOW SEE THE FOLLOWING ADDITIONAL PROBLEMS WORKED<BR>\n" );
document.write( "EARLIER.<BR>\n" );
document.write( "----------------------------------------------------------------------------------<BR>\n" );
document.write( "Matrices-and-determiminant/17820: Please help I have<BR>\n" );
document.write( "no idea how to do this problem: Use Cramer's Rule to<BR>\n" );
document.write( "solve each system.<BR>\n" );
document.write( "1. 2x+y=4<BR>\n" );
document.write( "3x-y=6 </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );