document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1190343>Question 1190343</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I am really having a hard time understanding solving using substitution \r<BR>\n" );
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document.write( "2x+3y=24<BR>\n" );
document.write( "-3x-5y=15\r<BR>\n" );
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document.write( "how do I solve those using substitution  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #821957 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=821957\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> with substitution, you solve for one of the variable in terms of the other variable in one of the equations and then you use that value to solve for the other variable in the other equation.\r<BR>\n" );
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document.write( "it's a way of reducing two unknown variable to one unknown variable.\r<BR>\n" );
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document.write( "a simpler example would be like this.\r<BR>\n" );
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document.write( "consider these two equations.\r<BR>\n" );
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document.write( "x + y = 1<BR>\n" );
document.write( "2x + 3y = 10\r<BR>\n" );
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document.write( "solve for x in the first equation to get:<BR>\n" );
document.write( "x = 1 - y\r<BR>\n" );
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document.write( "replace x in the second equation to get:<BR>\n" );
document.write( "2 * (1 - y) + 3y = 10<BR>\n" );
document.write( "simplify to get:<BR>\n" );
document.write( "2 - 2y + 3y = 10<BR>\n" );
document.write( "combine like terms to get:<BR>\n" );
document.write( "2 + y = 10<BR>\n" );
document.write( "solve for y to get:<BR>\n" );
document.write( "y = 10 - 2 = 8.\r<BR>\n" );
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document.write( "now that you know the value of y, solve for x in either equation to get:\r<BR>\n" );
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document.write( "x + y = 1 becomes x + 8 = 1 which becomes x = -7.\r<BR>\n" );
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document.write( "your values are x = -7 and y = 8\r<BR>\n" );
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document.write( "go back to the original equations and replace a with -7 and y with 8.<BR>\n" );
document.write( "you will get:\r<BR>\n" );
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document.write( "x + y = 1 becomes -7 + 8 = 1 which becomes 1 = 1 which is true.<BR>\n" );
document.write( "2x + 3y = 10 becomes -14 + 24 = 10 which becomes 10 = 10 which is true.<BR>\n" );
document.write( "this confirms the values of x and y are good because the same value of x and y make both equations true.\r<BR>\n" );
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document.write( "that was just an example.\r<BR>\n" );
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document.write( "your problem is a little more complicated because of the arithmetic involved but the concept is the same.\r<BR>\n" );
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document.write( "you are solving for one of the variables in terms of the other variable in one of the equation and you are using the result of that to solve for the other variable in the other equations.\r<BR>\n" );
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document.write( "your equations are:\r<BR>\n" );
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document.write( "2x + 3y = 24<BR>\n" );
document.write( "-3x - 5y = 15.\r<BR>\n" );
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document.write( "take the first equation and solve for y in terms of x.\r<BR>\n" );
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document.write( "you will get x = (24 - 3y) / 2.\r<BR>\n" );
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document.write( "replace x with that in the second equation.\r<BR>\n" );
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document.write( "you will get:\r<BR>\n" );
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document.write( "-3x - 5y = 15 becomes:<BR>\n" );
document.write( "-3 * [(24 - 3y) / 2] - 5y = 15<BR>\n" );
document.write( "multiply both sides of that equation by 2 to get:<BR>\n" );
document.write( "-3 * (24 - 3y) - 10y = 30<BR>\n" );
document.write( "simplify to get:<BR>\n" );
document.write( "-3 * 24 + 3 * 3y - 10y = 30<BR>\n" );
document.write( "simplify further to get:<BR>\n" );
document.write( "-72 + 9y - 10y = 30<BR>\n" );
document.write( "combine like terms to get:<BR>\n" );
document.write( "-72 - y = 30<BR>\n" );
document.write( "add 72 to both sides of the equation to get:<BR>\n" );
document.write( "-y = 102<BR>\n" );
document.write( "solve for y to get:<BR>\n" );
document.write( "y = -102\r<BR>\n" );
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document.write( "now that you know the value of y, you can use that value of y to solve for x in either of the original two equations.\r<BR>\n" );
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document.write( "2x + 3y = 24 becomes 2x + 3 * -102) = 24 which becomes 2x - 306 = 24 which becomes 2x = 330 which becomes x = 165.\r<BR>\n" );
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document.write( "-3x - 5y = 15 becomes -3x - 5 * -102 = 15 which becomes -3x + 510 = 15 which becomes -3x = -495.<BR>\n" );
document.write( "solve for x to get:<BR>\n" );
document.write( "x = -495 / -2 = 165.\r<BR>\n" );
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document.write( "your solution is that x = 165 and y = -102.\r<BR>\n" );
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document.write( "replace x and y with those values in the original equations and you will get:\r<BR>\n" );
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document.write( "2x+3y=24 becomes 2*165 - 3*102 = 24 which becomes 24 = 24 2 which is true.<BR>\n" );
document.write( "-3x-5y=15 becomes -3*165 +5*102 = 15 which becomes 15 = 15 which is also true.<BR>\n" );
document.write( "this confirm the solution is correct.\r<BR>\n" );
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document.write( "here's a reference on solving a system of equations simultaneously with substitution.\r<BR>\n" );
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document.write( "<a  rel=nofollow HREF= \"https://www.purplemath.com/modules/systlin4.htm\" target = \"_blank\">https://www.purplemath.com/modules/systlin4.htm</a>\r<BR>\n" );
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document.write( " </SPAN>\n" );
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