document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=23140>Question 23140</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I am having problems with solving equations.  The following are equations and it's methods that I don't seem to understand.  Thank you for your assistance!\r<BR>\n" );
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document.write( "Solve using the addition (elimination) method.<BR>\n" );
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document.write( "2x + 3y = -16<BR>\n" );
document.write( "5x-10y=30\r<BR>\n" );
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document.write( "Solve using the addition (elimination) method<BR>\n" );
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document.write( "2y - 8 = -2x - 8x<BR>\n" );
document.write( "8x - 3y = 31 + y\r<BR>\n" );
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document.write( "Solve using the substitution method<BR>\n" );
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document.write( "x = 3y + 7<BR>\n" );
document.write( "x = 2y - 1\r<BR>\n" );
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document.write( "Solve using the substitution method<BR>\n" );
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document.write( "4x + 3y = 0<BR>\n" );
document.write( "2x - y = 0<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 #11621 by </B><A HREF=/tutors/aboutme.mpl?userid=AnlytcPhil><B>AnlytcPhil(1806)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=AnlytcPhil><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=AnlytcPhil><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=11621\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "I am having problems with solving equations. The \r\n" );
document.write( "following are equations and it's methods that I don't \r\n" );
document.write( "seem to understand. Thank you for your assistance! \r\n" );
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document.write( "Solve using the addition (elimination) method.\r\n" );
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document.write( "They are already in standard form\r\n" );
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document.write( "Ax + By = C\r\n" );
document.write( "Dx + Ey = F\r\n" );
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document.write( "If they weren't in that form we'd have to get them \r\n" );
document.write( "that way, but they already are.\r\n" );
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document.write( "2x +  3y = -16\r\n" );
document.write( "5x - 10y =  30 \r\n" );
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document.write( "Look at the coefficients of x.  They are 2 and 5.  \r\n" );
document.write( "We want to cause new coefficients to occur where \r\n" );
document.write( "they are that will  be equal in absolute value but \r\n" );
document.write( "opposite in sign so they will cancel out.  The least \r\n" );
document.write( "common multiple of 2 and 5 is 10.  We can cause a \r\n" );
document.write( "10x to appear where the 2x is by multiplying the \r\n" );
document.write( "first equation through by 5.  We can cause a 10x to \r\n" );
document.write( "appear where the 5x is by multiplying the second \r\n" );
document.write( "equation through by 2.  But we will choose one of \r\n" );
document.write( "these multipliers to be negative (it doesn't matter \r\n" );
document.write( "which one) so that they will be opposite in sign and \r\n" );
document.write( "cancel.  So I will multiply the first equation by -5 \r\n" );
document.write( "and the second equation by +2.\r\n" );
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document.write( "-5[2x +  3y = -16]\r\n" );
document.write( " 2[5x - 10y =  30]\r\n" );
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document.write( "Multiplying through:\r\n" );
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document.write( " -10x - 15y = 80\r\n" );
document.write( "  10x - 20y = 60\r\n" );
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document.write( "Now we draw a line under them and add vertically\r\n" );
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document.write( " -10x - 15y = 80\r\n" );
document.write( "  10x - 20y = 60\r\n" );
document.write( " -----------------\r\n" );
document.write( "       -35y = 140\r\n" );
document.write( "          y = 140/(-35)\r\n" );
document.write( "          y = -4\r\n" );
document.write( "          \r\n" );
document.write( "2x +  3y = -16\r\n" );
document.write( "5x - 10y =  30 \r\n" );
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document.write( "Now look at the coefficients of y.  They are 3 and -10.\r\n" );
document.write( "We want to cause new coefficients to occur where they \r\n" );
document.write( "are that will  be equal in absolute value but opposite \r\n" );
document.write( "in sign so they will cancel out.  The least common \r\n" );
document.write( "multiple of 3 and 10 is 30.  We can cause a 30y to \r\n" );
document.write( "appear where the 3y is by multiplying the first equation\r\n" );
document.write( "through by 10.  We can cause a -30y to appear where the\r\n" );
document.write( "-10y is by multiplying the second equation through by 3.\r\n" );
document.write( "We don't need choose to them opposite in sign as we did\r\n" );
document.write( "before because they already are opposite in sign.  So I\r\n" );
document.write( "will multiply the first equation by 10 and the second \r\n" );
document.write( "equation by 3.\r\n" );
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document.write( "10[2x +  3y = -16]\r\n" );
document.write( " 3[5x - 10y =  30]\r\n" );
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document.write( "Multiplying through:\r\n" );
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document.write( "  20x + 30y = -160\r\n" );
document.write( "  15x - 30y =   90\r\n" );
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document.write( "Now we draw a line under them and add vertically\r\n" );
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document.write( "  20x + 30y = -160\r\n" );
document.write( "  15x - 30y =   90\r\n" );
document.write( " -----------------\r\n" );
document.write( "  35x       =  -70 \r\n" );
document.write( "          x = -70/35\r\n" );
document.write( "          y = -2\r\n" );
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document.write( "So the solution is (x,y) = (-4,-2)\r\n" );
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document.write( "<font size = 2></b>\r\n" );
document.write( "Solve using the addition (elimination) method\r\n" );
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document.write( "2y - 8 = -2x - 8x\r\n" );
document.write( "8x - 3y = 31 + y \r\n" );
document.write( "<font size = 3><b>\r\n" );
document.write( "Let's first get them like in stanbdard form like this:\r\n" );
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document.write( "Ax + By = C\r\n" );
document.write( "Dx + Ey = F\r\n" );
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document.write( "The first one:\r\n" );
document.write( "  2y - 8 = -2x - 8x\r\n" );
document.write( "  2y - 8 = -10x\r\n" );
document.write( "10x + 2y = 8\r\n" );
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document.write( "Notice that we can divide this equation through by 2. \r\n" );
document.write( "This isn't absolutely necessary by since we can it will\r\n" );
document.write( "make it easier so we divide through by 2 and get\r\n" );
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document.write( "  5x + y = 8\r\n" );
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document.write( "Now working with the second equation:\r\n" );
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document.write( "  8x - 3y = 31 + y \r\n" );
document.write( "  8x - 2y = 31\r\n" );
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document.write( "So our system in standard order\r\n" );
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document.write( "  5x +  y =  8\r\n" );
document.write( "  8x - 2y = 31\r\n" );
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document.write( "We choose one of the equations to solve for one of the\r\n" );
document.write( "letters.  The easiest choice we can make is to pick the\r\n" );
document.write( "first equation to solve for y.  (Whenever possible choose\r\n" );
document.write( "a letter that has a 1 or -1 coefficient, since that makes\r\n" );
document.write( "it easier).\r\n" );
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document.write( "Solving the first equation for y\r\n" );
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document.write( "  5x + y = 8\r\n" );
document.write( "       y = 8 - 5x\r\n" );
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document.write( "Now go to the OTHER equation\r\n" );
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document.write( " 8x - 2y = 31\r\n" );
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document.write( "Substitute 8 - 5x for y in that: \r\n" );
document.write( "       \r\n" );
document.write( " 8x - 2(8 - 5x) = 31  \r\n" );
document.write( "  8x - 16 + 10x = 31\r\n" );
document.write( "            18x = 47 \r\n" );
document.write( "              x = 47/18\r\n" );
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document.write( "Now we substitute 47/18 for x in the equation solved \r\n" );
document.write( "for y:\r\n" );
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document.write( "y = 8 - 5x\r\n" );
document.write( "y = 8 - 5(47/18)\r\n" );
document.write( "y = 8 - 235/18\r\n" );
document.write( "y = 144/18 - 235/18\r\n" );
document.write( "y = -91/18\r\n" );
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document.write( "So the solution is (x,y) = (47/18,-91/18)\r\n" );
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document.write( "Solve using the substitution method\r\n" );
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document.write( "x = 3y + 7\r\n" );
document.write( "x = 2y - 1 \r\n" );
document.write( "<font size = 3><b>\r\n" );
document.write( "These are already solved for on letter, so we substitute\r\n" );
document.write( "3y + 7 for x in the second equation:\r\n" );
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document.write( "     x = 2y - 1\r\n" );
document.write( "3y + 7 = 2y - 1\r\n" );
document.write( "     y = -8\r\n" );
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document.write( "Now substitute -8 for y in either one of the original\r\n" );
document.write( "equations, say, the first:\r\n" );
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document.write( "x = 3y + 7\r\n" );
document.write( "x = 3(-8) + 7\r\n" );
document.write( "x = -24 + 7\r\n" );
document.write( "x = -17\r\n" );
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document.write( "So the solution is (x, y) = (-17,-8) \r\n" );
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document.write( "<font size = 2></b>\r\n" );
document.write( "Solve using the substitution method\r\n" );
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document.write( "4x + 3y = 0\r\n" );
document.write( "2x -  y = 0\r\n" );
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document.write( "Choose the second one to solve for y, since that's the\r\n" );
document.write( "simpler.\r\n" );
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document.write( "2x - y = 0\r\n" );
document.write( "    -y = -2x\r\n" );
document.write( "     y = 2x\r\n" );
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document.write( "Substitute 2x for y in the OTHER equation\r\n" );
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document.write( "   4x + 3y = 0 \r\n" );
document.write( "4x + 3(2x) = 0\r\n" );
document.write( "   4x + 6x = 0\r\n" );
document.write( "       10x = 0\r\n" );
document.write( "         x = 0/10\r\n" );
document.write( "         x = 0\r\n" );
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document.write( "Substitute 0 for x in the equation y = 2x\r\n" );
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document.write( "         y = 2x\r\n" );
document.write( "         y = 2(0)\r\n" );
document.write( "         y = 0\r\n" );
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document.write( "So the solution is (x,y) = (0,0) \r\n" );
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document.write( "Edwin\r\n" );
document.write( "AnlytcPhil@aol.com</PRE>  </SPAN>\n" );
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