document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=131812>Question 131812</A>: <I><SPAN STYLE=\"word-break: break-all;\"> how could i solve this linear equation using a combination method<BR>\n" );
document.write( "                  4x-3y=11<BR>\n" );
document.write( "                  3x+2y=-13<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #96276 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=96276\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> You are not solving a linear equation ... you are solving a set of linear equations. <BR>\n" );
document.write( "What you are doing is trying to find a single point (x, y) that will satisfy both of the equations.<BR>\n" );
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document.write( "There are several ways this can be done:<BR>\n" );
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document.write( "You could graph both equations and find the (x, y) point where the two graphs intersect.<BR>\n" );
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document.write( "You could solve one equation for one of the variables and substitute that into the other equation<BR>\n" );
document.write( "to eliminate one of the two variables in it. Then solve for the remaining variable.<BR>\n" );
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document.write( "You could add or subtract the two equations in such a way as to eliminate one of the variables<BR>\n" );
document.write( "and solve for the remaining variable.<BR>\n" );
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document.write( "You were given the set of equations:<BR>\n" );
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document.write( "4x-3y=11<BR>\n" );
document.write( "3x+2y=-13<BR>\n" );
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document.write( "Let's try combining the two equations by addition or subtraction to eliminate one of the<BR>\n" );
document.write( "variables. The first goal will be for us to get a term in each equation to be the same<BR>\n" );
document.write( "size. Let's try to get the y terms to be the same in both equations. We can do that by<BR>\n" );
document.write( "multiplying the top equation (both sides and all terms) by 2 and the bottom equation<BR>\n" );
document.write( "(both sides and all terms by 3). First multiplying the top equation by 2 converts it<BR>\n" );
document.write( "to:<BR>\n" );
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document.write( "8x - 6y = 22<BR>\n" );
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document.write( "Next multiplying the bottom equation by 3 converts it to:<BR>\n" );
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document.write( "9x + 6y = - 39<BR>\n" );
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document.write( "So our set of the two equations is now:<BR>\n" );
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document.write( "8x - 6y = 22<BR>\n" );
document.write( "9x + 6y = -39<BR>\n" );
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document.write( "Notice now what happens if we add these two equations together in vertical columns.<BR>\n" );
document.write( "The -6y and the +6y will cancel each other out ... we made it that way.<BR>\n" );
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document.write( "If we add the two equations in vertical columns, the 8x and the 9x add to 17x, the two<BR>\n" );
document.write( "y terms cancel each other as we already noted, and the 22 and the -39 add to -17. So the<BR>\n" );
document.write( "problem simplifies to:<BR>\n" );
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document.write( "8x - 6y = 22<BR>\n" );
document.write( "9x + 6y = -39<BR>\n" );
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document.write( "17x = -17<BR>\n" );
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document.write( "Solve for x by dividing both sides of the \"added\" equation by 17 to get:<BR>\n" );
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document.write( "x = -17/17 = -1<BR>\n" );
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document.write( "Now we know that the x value of the common solution is -1. We can go back to any of the <BR>\n" );
document.write( "equations and substitute -1 for x and solve for the corresponding value of y. For example,<BR>\n" );
document.write( "let's go back to the original equation:<BR>\n" );
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document.write( "4x-3y=11<BR>\n" );
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document.write( "If we substitute -1 for x this equation becomes:<BR>\n" );
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document.write( "4(-1) - 3y = 11<BR>\n" );
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document.write( "Multiply out the 4 times -1 and get -4 ... making the equation:<BR>\n" );
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document.write( "-4 - 3y = 11<BR>\n" );
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document.write( "Get rid of the -4 on the left side by adding +4 to both sides and the equation reduces to:<BR>\n" );
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document.write( "-3y = 15<BR>\n" );
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document.write( "Solve for y by dividing both sides by -3 to get:<BR>\n" );
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document.write( "y = 15/-3 = -5<BR>\n" );
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document.write( "You now have found the x and the corresponding y values that satisfy both equations.<BR>\n" );
document.write( "These values are x = -1 and y = -5.<BR>\n" );
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document.write( "This tells you several things. If you graph the two equations their line graphs will cross<BR>\n" );
document.write( "at the point (-1, -5). Also, if you substitute -1 for x and -5 for y into both of the<BR>\n" );
document.write( "original equations, the equations should balance. Let's do that substitution into both<BR>\n" );
document.write( "of the original equations, just to check our answer:<BR>\n" );
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document.write( "First original equation:<BR>\n" );
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document.write( "4x-3y=11<BR>\n" );
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document.write( "Substitute -1 for x and -5 for y and you have:<BR>\n" );
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document.write( "4(-1) - 3(-5) = 11 <=== multiply out the left side to get:<BR>\n" );
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document.write( "-4 + 15 = 11<BR>\n" );
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document.write( "Combining the two terms on the left side results in the equation reducing to 11= 11 and<BR>\n" );
document.write( "since both sides are equal, this checks.<BR>\n" );
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document.write( "Now do the same thing for the second of the original two equations:<BR>\n" );
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document.write( "3x+2y=-13<BR>\n" );
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document.write( "Substitute -1 for x and -5 for y to get:<BR>\n" );
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document.write( "3(-1) + 2(-5) = -13 <==== multiply out the left side to get:<BR>\n" );
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document.write( "-3 - 10 = -13<BR>\n" );
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document.write( "When you combine the two terms on the left side the equation becomes -13 = -13 and since<BR>\n" );
document.write( "both sides are equal this second check also works.<BR>\n" );
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document.write( "Therefore, we know the answer x = -1 and y = -5 is the common solution for both equations.<BR>\n" );
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document.write( "Hope this helps you to understand one way of doing problems such as these.<BR>\n" );
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