document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=230125>Question 230125</A>: <I><SPAN STYLE=\"word-break: break-all;\"> please help me understand the difference between substitution and elimination method. use substitution to solve this system\r<BR>\n" );
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document.write( "3x-y=6<BR>\n" );
document.write( "-4x+2y=-8 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #170461 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=170461\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> 3x - y = 6 (equation 1)<BR>\n" );
document.write( "-4x + 2y = -8 (equation 2)\r<BR>\n" );
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document.write( "Solve by substitution.\r<BR>\n" );
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document.write( "Take equation 1 and solve for y.\r<BR>\n" );
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document.write( "3x - y = 6 (equation 1)<BR>\n" );
document.write( "add y to both sides of the equation to get:<BR>\n" );
document.write( "y + 6 = 3x<BR>\n" );
document.write( "subtract 6 from both sides of the eqution to get:<BR>\n" );
document.write( "y = 3x - 6\r<BR>\n" );
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document.write( "You have now solved for y in terms of x in equation 1.\r<BR>\n" );
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document.write( "Take equation 2 and replace y with 3x - 6.  This is the substitution part.\r<BR>\n" );
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document.write( "-4x + 2y = -8 (equation 2)<BR>\n" );
document.write( "Replace y with value of y calculated from equation 1.<BR>\n" );
document.write( "-4x + (2 * (3x-6)) = -8<BR>\n" );
document.write( "Remove parentheses.<BR>\n" );
document.write( "-4x + 2*3x - 2*6 = -8<BR>\n" );
document.write( "Simplify.<BR>\n" );
document.write( "-4x + 6x - 12 = -8<BR>\n" );
document.write( "Combine like terms.<BR>\n" );
document.write( "2x - 12 = -8<BR>\n" );
document.write( "Add 12 to both sides of the equation.<BR>\n" );
document.write( "2x = -8 + 12<BR>\n" );
document.write( "Simplify and combine like terms.<BR>\n" );
document.write( "2x = 4<BR>\n" );
document.write( "Divide both sides of the equation by 2.<BR>\n" );
document.write( "x = 4/2 = 2\r<BR>\n" );
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document.write( "You have now solved for x in equation 2.\r<BR>\n" );
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document.write( "Take the value of x and replace x with it in equation 1 to solve for y.\r<BR>\n" );
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document.write( "3x - y = 6 (equation 1)<BR>\n" );
document.write( "Replace x with 2.<BR>\n" );
document.write( "3*2 - y = 6<BR>\n" );
document.write( "Simplify.<BR>\n" );
document.write( "6 - y = 6<BR>\n" );
document.write( "Subtract 6 from both sides of the equation.<BR>\n" );
document.write( "-y = 0<BR>\n" );
document.write( "Multiply both sides of the equation by -1.<BR>\n" );
document.write( "y = 0\r<BR>\n" );
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document.write( "You have now solved for y in equation 1.\r<BR>\n" );
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document.write( "The values that you calculated for both equations are:\r<BR>\n" );
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document.write( "x = 2<BR>\n" );
document.write( "y = 0\r<BR>\n" );
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document.write( "Replace x and y in both original equations to see if these values will solve both equations at the same time (simultaneously).\r<BR>\n" );
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document.write( "3x - y = 6 (equation 1)<BR>\n" );
document.write( "-4x + 2y = -8 (equation 2)\r<BR>\n" );
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document.write( "Replace x with 2 and y with 0 in both equations.\r<BR>\n" );
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document.write( "3x - y = 6 in equation 1 becomes 6 - 0 = 6 becomes 6 = 6 which is true.\r<BR>\n" );
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document.write( "-4x + 2y = -8 in equation 2 becomes -8 = 0 = -8 becomes -8 = -8 which is true.\r<BR>\n" );
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document.write( "Both equations are true indicating the values for x and y that you calculated will solve both equations simultaneously.\r<BR>\n" );
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document.write( "x = 2 and y = 0 are your answers for solving these two equations simultaneously.\r<BR>\n" );
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document.write( "You did it through the process of substitution.\r<BR>\n" );
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document.write( "Now we'll do elimination.\r<BR>\n" );
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document.write( "Same two equations are:\r<BR>\n" );
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document.write( "3x - y = 6 (equation 1)<BR>\n" );
document.write( "-4x + 2y = -8 (equation 2)\r<BR>\n" );
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document.write( "Object is to manipulate each equation by multiplying or dividing as necessary (mostly multiplying) so that one of the variables will have a common coefficient in both equations.  When you add or subtract one equation from the other, the variable with the common coefficients will cancel out and you will be left with one equation in one unknown variable that you can solve.\r<BR>\n" );
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document.write( "We will multiply equation 1 by 2.\r<BR>\n" );
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document.write( "6x - 2y = 12 (equation 1 multiplied by 2)<BR>\n" );
document.write( "-4x + 2y = -8 (equation 2)\r<BR>\n" );
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document.write( "Because the variable y in each equation has the same coefficients and the signs are different, we'll add equation 2 to equation 1 in order to eliminate the y variable.\r<BR>\n" );
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document.write( "6x - 2y = 12 (equation 1 multiplied by 2)<BR>\n" );
document.write( "plus:<BR>\n" );
document.write( "-4x + 2y = -8 (equation 2)<BR>\n" );
document.write( "equals:<BR>\n" );
document.write( "2x = 4<BR>\n" );
document.write( "Divide both sides of this equation by 2.<BR>\n" );
document.write( "x = 2\r<BR>\n" );
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document.write( "We have solved for x.\r<BR>\n" );
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document.write( "This was done through the process of elimination.\r<BR>\n" );
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document.write( "Now we go back to equation 1 and replace x with 2 and solve for y.\r<BR>\n" );
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document.write( "These are the same steps you did in the substitution part.  Once you get to this point, the remaining operations are the same regardless if you got here through substition or elimination.\r<BR>\n" );
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document.write( "3x - y = 6 (equation 1)<BR>\n" );
document.write( "Replace x with 2.<BR>\n" );
document.write( "3*2 - y = 6<BR>\n" );
document.write( "simplify.<BR>\n" );
document.write( "6 - y = 6<BR>\n" );
document.write( "Subtract 6 from both sides of the equation and add y to both sides of the equation.<BR>\n" );
document.write( "y = 0\r<BR>\n" );
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document.write( "We have solved for y in equation 1.\r<BR>\n" );
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document.write( "We have:\r<BR>\n" );
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document.write( "x = 2<BR>\n" );
document.write( "y = 0\r<BR>\n" );
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document.write( "Now we confirm in all three equations and we're done.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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