document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=112638>Question 112638</A>: <I><SPAN STYLE=\"word-break: break-all;\"> in solving systems of linear equations how do you know whether to eliminate the x's or y's? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #82119 by </B><A HREF=/tutors/aboutme.mpl?userid=bucky><B>bucky(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=bucky><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=82119\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> It doesn't make any difference which one you eliminate ... the answers will turn out to be the same.<BR>\n" );
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document.write( "Just eliminate the one that seems to be the easiest to do, whether it is x or it is y.<BR>\n" );
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document.write( "For example, take the two linear equations:<BR>\n" );
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document.write( "+4x + 3y = +29<BR>\n" );
document.write( "+2x + 6y = +28<BR>\n" );
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document.write( "Let's eliminate the x column by multiplying the bottom equation (all terms on both sides) by -2.<BR>\n" );
document.write( "This multiplication changes the bottom equation and the two equations are then:<BR>\n" );
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document.write( "+4x + 3y = +29<BR>\n" );
document.write( "-4x -12y = -56<BR>\n" );
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document.write( "Add the two equations in columns to eliminate the x column and you end up with:<BR>\n" );
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document.write( "-9y = -27<BR>\n" );
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document.write( "Solve for y by dividing both sides of this equation by -9 and you get:<BR>\n" );
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document.write( "y = -27/-9 = +3<BR>\n" );
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document.write( "Take this value of y and substitute it into one of the two original equations to solve for x.<BR>\n" );
document.write( "Let's substitute it into the equation:<BR>\n" );
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document.write( "+4x + 3y = +29<BR>\n" );
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document.write( "Replacing y by +3 changes this equation to:<BR>\n" );
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document.write( "+4x + 3*3 = +29<BR>\n" );
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document.write( "Do the multiplication on the left side to reduce the equation to:<BR>\n" );
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document.write( "+4x + 9 = +29<BR>\n" );
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document.write( "Get rid of the 9 on the left side by subtracting 9 from both sides to get:<BR>\n" );
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document.write( "4x = 20<BR>\n" );
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document.write( "Solve for x by dividing both sides by 4 and you have:<BR>\n" );
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document.write( "x = 20/4 = +5<BR>\n" );
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document.write( "So by eliminating the x terms we got an answer of y = 3 and x = 5.<BR>\n" );
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document.write( "Now let's return to the original set of equations and solve it by eliminating the y column.<BR>\n" );
document.write( "Start with:<BR>\n" );
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document.write( "+4x + 3y = +29<BR>\n" );
document.write( "+2x + 6y = +28<BR>\n" );
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document.write( "Multiply the top equation (all terms on both sides) by -2 to convert the equation set<BR>\n" );
document.write( "to:<BR>\n" );
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document.write( "-8x - 6y = -58<BR>\n" );
document.write( "+2x + 6y = +28<BR>\n" );
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document.write( "Add the two equations vertically to eliminate the y column and get:<BR>\n" );
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document.write( "-6x = -30<BR>\n" );
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document.write( "Solve for x by dividing both sides by -6 and you get:<BR>\n" );
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document.write( "x = -30/-6 = +5<BR>\n" );
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document.write( "Return to one of the two original equations and substitute +5 for x to solve for y. Let's <BR>\n" );
document.write( "use the bottom equation this time, although we could also use the top one again.<BR>\n" );
document.write( "The bottom<BR>\n" );
document.write( "equation is:<BR>\n" );
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document.write( "+2x + 6y = +28<BR>\n" );
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document.write( "Substitute 5 for x and it becomes:<BR>\n" );
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document.write( "2*5 + 6y = +28<BR>\n" );
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document.write( "Multiply on the left side:<BR>\n" );
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document.write( "+10 + 6y = +28<BR>\n" );
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document.write( "Get rid of the +10 on the left side by subtracting +10 from both sides and you have:<BR>\n" );
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document.write( "6y = 18<BR>\n" );
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document.write( "Solve for y by dividing both sides by 6 and the result is:<BR>\n" );
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document.write( "y = 18/6 = 3<BR>\n" );
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document.write( "So when we eliminated the x column we got y = 3 and x = 5 as the answer. And when we<BR>\n" );
document.write( "eliminated the y column we got x = 5 and y = 3 .... the answers are the same. This <BR>\n" );
document.write( "example shows you that it doesn't really make any difference which variable you choose to<BR>\n" );
document.write( "eliminate ... the results will be the same in either case.<BR>\n" );
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document.write( "Hope this helps you to understand the situation when using variable elimination to find<BR>\n" );
document.write( "the common solution for a pair of linear equations.<BR>\n" );
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