document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1166180>Question 1166180</A>: <I><SPAN STYLE=\"word-break: break-all;\"> please can help me \r<BR>\n" );
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document.write( "Solve the following linear equations using the Elimination method and The Substitution method<BR>\n" );
document.write( "1) X + 3 Y = 2<BR>\n" );
document.write( "2X + 5 Y = 3\r<BR>\n" );
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document.write( "2) 3X + 4 Y = 12<BR>\n" );
document.write( "6X + 2 Y = 6<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 #790675 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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/cgi-bin/show-face.mpl?user=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=790675\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the substitution method uses one of the equations to solve for a variable in the other equation, the aim being to reduce the number of unknowns to 1 in other equation.\r<BR>\n" );
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document.write( "the elimination method does the same thing, only it multiplies one or both of the equations so that one of the variables cancels out when you add or subtract one of the equations to or from the other.\r<BR>\n" );
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document.write( "once you have found what you think is your solution, you need to confirm it by replacing the variables in the original equations with the solution to confirm that the solution is correct.\r<BR>\n" );
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document.write( "substitution method comes first.\r<BR>\n" );
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document.write( "1) X + 3Y = 2 and 2X + 5Y = 3\r<BR>\n" );
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document.write( "in the equation of x + 3y = 2, solve for x to get:<BR>\n" );
document.write( "x = 2 - 3y.\r<BR>\n" );
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document.write( "in the equation of 2x + 5y = 3, replace x with 2 - 3y to get:<BR>\n" );
document.write( "2 * (2 - 3y) + 5y = 3.<BR>\n" );
document.write( "simplify to get:<BR>\n" );
document.write( "4 - 6y + 5y = 3.<BR>\n" );
document.write( "combine like terms and subtract 4 from both sides of the equation to get:<BR>\n" );
document.write( "-y = -1<BR>\n" );
document.write( "solve for y to get:<BR>\n" );
document.write( "y = 1\r<BR>\n" );
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document.write( "when y = 1, x + 3y = 2 becomes x + 3 = 2 which becomes x = -1.\r<BR>\n" );
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document.write( "your solution is x = -1 and y = 1.\r<BR>\n" );
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document.write( "to confirm, replace x and y in the original equations to see if they are true.\r<BR>\n" );
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document.write( "the original equations are X + 3Y = 2 and 2X + 5Y = 3<BR>\n" );
document.write( "they become -1 + 3 = 2 which is true and -2 + 5 = 3 which is also true.<BR>\n" );
document.write( "the solution is confirmed to be good.\r<BR>\n" );
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document.write( "2) 3X + 4Y = 12 and 6X + 2Y = 6\r<BR>\n" );
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document.write( "in the equation of 6x + 2y = 6, solve for y as follows:<BR>\n" );
document.write( "start with 6x + 2y = 6<BR>\n" );
document.write( "subtract 6x from both sides and divide both sides by 2 to get:<BR>\n" );
document.write( "y = (6 - 6x) / 2 = (3 - 3x).\r<BR>\n" );
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document.write( "in the equation of 3x + 4y = 12, replace y with (3 - 3x) to get:<BR>\n" );
document.write( "3x + 4 * (3 - 3x) = 12<BR>\n" );
document.write( "simplify to get:<BR>\n" );
document.write( "3x + 12 - 12x = 12<BR>\n" );
document.write( "combine like terms and subtract 12 from both sides to get:<BR>\n" );
document.write( "-9x = 0<BR>\n" );
document.write( "solve for x to get:<BR>\n" );
document.write( "x = 0\r<BR>\n" );
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document.write( "in the equation of y = (3 - 3x), replace x with 0 to get y = 3\r<BR>\n" );
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document.write( "your solution is that x = 0 and y = 3.\r<BR>\n" );
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document.write( "to confirm the solution is good, replace x and y with 0 and 3 in the original equations to see if they are true.\r<BR>\n" );
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document.write( "the original equations are 3X + 4Y = 12 and 6X + 2Y = 6.<BR>\n" );
document.write( "3x + 4y = 12 becomes 3 * 0 + 4 * 3 = 12 which becomes 12 = 12 which is true.<BR>\n" );
document.write( "6x + 2y = 6 becomes 6 * 0 + 2 * 3 = 6 which becomes 6 = 6 which is also true.<BR>\n" );
document.write( "the solution is confirmed to be good.\r<BR>\n" );
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document.write( "problem 1 solution was x = -1 and y = 1.<BR>\n" );
document.write( "problem 2 solution was x = 0 and y = 3.\r<BR>\n" );
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document.write( "now we'll do the same problems using the elimination method. \r<BR>\n" );
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document.write( "1) X + 3Y = 2 and 2X + 5Y = 3\r<BR>\n" );
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document.write( "multiply both sides of the first equation by 2 and leave the second equation as is to get:\r<BR>\n" );
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document.write( "2x + 6y = 4<BR>\n" );
document.write( "2x + 5y = 3\r<BR>\n" );
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document.write( "subtract the second equation from the first to get:\r<BR>\n" );
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document.write( "y = 1\r<BR>\n" );
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document.write( "when y = 1, x + 3y = 2 becomes x + 3 = 2 which becomes x = -1.\r<BR>\n" );
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document.write( "your solution is x = -1 and y = 1, same as the solution using the substitution method, therefore confirmed to be good.\r<BR>\n" );
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document.write( "2) 3X + 4Y = 12 and 6X + 2Y = 6\r<BR>\n" );
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document.write( "multiply both sides of the first equation by 2 and leave the second equation as is to get:\r<BR>\n" );
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document.write( "6x + 8y = 24<BR>\n" );
document.write( "6x + 2y = 6\r<BR>\n" );
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document.write( "subtract the second equation from the first to get:<BR>\n" );
document.write( "6y = 18.<BR>\n" );
document.write( "solve for y to get:<BR>\n" );
document.write( "y = 3\r<BR>\n" );
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document.write( "when y = 3, 6x + 2y = 6 becomes:<BR>\n" );
document.write( "6x + 6 = 6<BR>\n" );
document.write( "solve for x to get:<BR>\n" );
document.write( "x = 0\r<BR>\n" );
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document.write( "your solution is that x = 0 and y = 3, same as the solution using the substitution method, therefore confirmed to be good.\r<BR>\n" );
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document.write( "both systems of equations have been solved using the substitution method and the elimination method, yielding the same solutions using either method.\r<BR>\n" );
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