document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=21095>Question 21095</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Is there a way to easily solve matrices, 3 system equations, and Cramer's rule <BR>\n" );
document.write( "problems?\r<BR>\n" );
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document.write( "What are the short cuts or tricks to solving the above problems? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #11865 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=11865\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <BR>\n" );
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document.write( "2x+y=4<BR>\n" );
document.write( "3x-y=6<BR>\n" );
document.write( "make a deteminant with coefficients of x (2,3)and y(1,-1) in the 2 eqns.call it C.(Actually for a determinant as you know ,the numbers are contained in vertical bars at either end like |xx|,but in the following the bars are omitted due to difficulty in depiction.you may assume the bars are present)<BR>\n" );
document.write( "C=matrix(2,2,2,1,3,-1)=2*(-1)-(1*3)=-5<BR>\n" );
document.write( "..now use the constants (4,6)to replace coefficients of x(2,3) in the above determinant C...call it CX..<BR>\n" );
document.write( "CX=matrix(2,2,4,1,6,-1)=4*(-1)-1*6=-4-6=-10<BR>\n" );
document.write( "..now use the constants (4,6)to replace coefficients of y(1,-1) in the above determinant C...call it CY..<BR>\n" );
document.write( "CY=matrix(2,2,2,4,3,6)=2*6-3*4=12=12=0<BR>\n" );
document.write( "..now cramers rule says that<BR>\n" );
document.write( "(x/CX)=(y/CY)=(1/C)..so we get<BR>\n" );
document.write( "x/(-10)=y/0=1/-5<BR>\n" );
document.write( "x=-10/-5=10/5=2<BR>\n" );
document.write( "y=0/-5=0<BR>\n" );
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document.write( "so using the above method you can do the next problem ..here due to presence of 3 variables you will get 3rd.order determinants...4 in all...namely C,CX,CY and CZ,the last formula also extends to include z ,<BR>\n" );
document.write( "(x/CX)=(y/CY)=(z/CZ)=(1/C)..<BR>\n" );
document.write( "but the procedure is same ..<BR>\n" );
document.write( "2x+3y+ z= 5<BR>\n" );
document.write( "x+y-2z= -2<BR>\n" );
document.write( "-3x +z=-7 ...<BR>\n" );
document.write( "...just to give you the idea<BR>\n" );
document.write( "C=matrix(3,3,2,3,1,1,1,-2,-3,0,1)..and<BR>\n" );
document.write( "CZ=matrix(3,3,2,3,5,1,1,-2,-3,0,7)..etc..hope you can work out the rest\r<BR>\n" );
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document.write( "Kramers rule is used to solve simultaneous equations.In the working given below <BR>\n" );
document.write( "let me use the notation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%282%2C2%2C1%2C2%2C3%2C4%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%282%2C2%2C1%2C2%2C3%2C4%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >to represent  the determinant.However,in actual practice we should use |xxx|to indicate a determinant ,while (xxx)is used to represent a matrix.I shall check with the network authorities on how to represent determinants in their software. <BR>\n" );
document.write( " In the present case there are 3 equations to solve for 3 unknowns.<BR>\n" );
document.write( "The solution is given by Kramer's rule as follows. <BR>\n" );
document.write( " x/<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%283%2C3%2C-2%2C3%2C1%2C-5%2C5%2C1%2C6%2C2%2C3%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%283%2C3%2C-2%2C3%2C1%2C-5%2C5%2C1%2C6%2C2%2C3%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >=<BR>\n" );
document.write( " y/<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%283%2C3%2C1%2C-2%2C1%2C2%2C-5%2C1%2C1%2C6%2C3%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%283%2C3%2C1%2C-2%2C1%2C2%2C-5%2C1%2C1%2C6%2C3%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >=<BR>\n" );
document.write( " z/<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%283%2C3%2C1%2C3%2C-2%2C2%2C5%2C-5%2C1%2C2%2C6%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%283%2C3%2C1%2C3%2C-2%2C2%2C5%2C-5%2C1%2C2%2C6%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >=<BR>\n" );
document.write( " 1/<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%283%2C3%2C1%2C3%2C1%2C2%2C5%2C1%2C1%2C2%2C3%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%283%2C3%2C1%2C3%2C1%2C2%2C5%2C1%2C1%2C2%2C3%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "evaluating the determinants we get <BR>\n" );
document.write( " x/(-3)=y/6=z/(-9)=1/(-3)<BR>\n" );
document.write( "  hence x=1<BR>\n" );
document.write( "        y=-2<BR>\n" );
document.write( "        z=3\r<BR>\n" );
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document.write( "I have working on this problem for some time but I am still having a hard time working this one:<BR>\n" );
document.write( "I am suppose to solve this system using elimination method<BR>\n" );
document.write( "10x+6y+z=7 (1)<BR>\n" );
document.write( "5x-9y-2z=3 (2)<BR>\n" );
document.write( "15x-12y+2z=-5 (3) <BR>\n" );
document.write( "20x-21y=-2 (4) <BR>\n" );
document.write( "Please show me how to work this problem from this point.<BR>\n" );
document.write( "Thank you <BR>\n" );
document.write( "Good you have proceeded correctly and infact on the way to solving the problem by your self..you only need a little guidance on the path you should follow to solve the problem..o.k. ..let us see you have added equations 2 and 3 to get equation 3 ,which has accomplished elimination of one unknown z . The basic procedure is , if we start with 3 equations in 3 unknowns ,we try to eliminate one unknown taking one pair of equations at a time to get 2 new equations in 2 unknowns only.Then we take those 2 new equations to eliminate one another unknown to get one more new equation , but this time with one unknown only.This we can easily solve to find the unknown.Now , we travel backwards along the same path as we travelled to find the 2 other unknowns one after another by substituting the known values every time.Let us illustrate the procedure now with this example.Now that you have already got one new equation 4 from 2 and 3 to eliminate z., let us take equations 1 and 2 to eliminate the same unknown z.For this we observe the coefficients of z in the two equations which are 1 and -2 respectively.So we multiply equation 1 with 2 and add it to equation 2.<BR>\n" );
document.write( "Eqn.1 * 2 gives us ...20x+12y+2z=14 .....(5)<BR>\n" );
document.write( "Eqn.2 is .............5x-9y-2z = 3........(6)<BR>\n" );
document.write( "Eqn.5 + Eqn.6 gives us .....25x+3y = 17....(7)<BR>\n" );
document.write( "but from Eqn.4 we have .....20x-21y=-2......(4)..proceeding on the same basis ,we eliminate y from these 2 equations.<BR>\n" );
document.write( "Eqn.7 * 7 gives us .........175x+21y=119....(8)<BR>\n" );
document.write( "Eqn.8 + Eqn.4 gives us .....195x=117 ..or x= 117/195 = 39/65 = 3/5.....now substitute this value of x in eqn.4 to get y<BR>\n" );
document.write( "y=(20*(3/5)+2)/21=14/21=2/3…….now substitute these values of x and y in eqn.1 to get z.<BR>\n" );
document.write( "z=(7-10*(3/5)-6*(2/3))=-3………….. As a check ,you can substitute these values of x,y,and z in the 3 given equations to <BR>\n" );
document.write( "verify that your answer is correct.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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