document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1199023>Question 1199023</A>: <I><SPAN STYLE=\"word-break: break-all;\"> solve the following systems using the elimination method. Write your complete solutions. \r<BR>\n" );
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document.write( "1.  x - 2y = 5 and 2x + 3y = -32<BR>\n" );
document.write( "2.  3x + 4y = 3 and 2x - 3y = 19<BR>\n" );
document.write( "3.  3x + 5y = -14 and 2x - y = -18 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #832707 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=832707\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "The rule on this website is that you need to post one problem at a time (or one problem per post). <BR>\n" );
document.write( "I'll do problem 1 to get you started.\r<BR>\n" );
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document.write( "The system we have is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28x-2y=5%2C2x%2B3y=-32%29 ALIGN=MIDDLE  ALT=\"system%28x-2y=5%2C2x%2B3y=-32%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "If we were to add the equations straight down, then:<BR>\n" );
document.write( "x+2x = 3x<BR>\n" );
document.write( "-2y+3y = 1y = y<BR>\n" );
document.write( "None of the variables cancel. <BR>\n" );
document.write( "Same goes if we were to subtract the equations.\r<BR>\n" );
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document.write( "What we need is to have the coefficients of either x or y match up so they cancel.\r<BR>\n" );
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document.write( "Let's say we doubled everything in the 1st equation<BR>\n" );
document.write( "We'd go from x-2y = 5 to 2x-4y = 10\r<BR>\n" );
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document.write( "Now we have this updated equivalent system<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%282x-4y=10%2C2x%2B3y=-32%29 ALIGN=MIDDLE  ALT=\"system%282x-4y=10%2C2x%2B3y=-32%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "We can subtract the equations straight down<BR>\n" );
document.write( "2x-2x = 0x = 0, the x terms go away<BR>\n" );
document.write( "-4x-3y = -7y<BR>\n" );
document.write( "10-(-32) = 10+32 = 42\r<BR>\n" );
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document.write( "We end up with the equation -7y = 42 which solves to y = -6.\r<BR>\n" );
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document.write( "Then use that y value to find x.<BR>\n" );
document.write( "Pick either equation with x and y.<BR>\n" );
document.write( "Let's say we chose the 1st equation.<BR>\n" );
document.write( "x - 2y = 5<BR>\n" );
document.write( "x - 2(-6) = 5<BR>\n" );
document.write( "x + 12 = 5<BR>\n" );
document.write( "x = 5-12<BR>\n" );
document.write( "x = -7<BR>\n" );
document.write( "Or you can pick on the 2nd equation.<BR>\n" );
document.write( "2x+3y = -32<BR>\n" );
document.write( "2x+3(-6) = -32<BR>\n" );
document.write( "2x-18 = -32<BR>\n" );
document.write( "2x = -32+18<BR>\n" );
document.write( "2x = -14<BR>\n" );
document.write( "x = -14/2<BR>\n" );
document.write( "x = -7<BR>\n" );
document.write( "Either way you should get the same x value. <BR>\n" );
document.write( "This helps confirm you have the correct solution.\r<BR>\n" );
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document.write( "A visual way to confirm the answer is to graph each equation onto the same xy grid.<BR>\n" );
document.write( "I recommend either Desmos or GeoGebra as two graphing options.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28400%2C400%2C-10%2C10%2C-10%2C10%2C-100%2C%285-x%29%2F%28-2%29%2C%28-32-2x%29%2F%283%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "x - 2y = 5 in green and 2x + 3y = -32 in blue<BR>\n" );
document.write( "The two lines intersect at (-7, -6)\r<BR>\n" );
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document.write( "Or you could plug each coordinate of (-7,-6) back into the original equations. <BR>\n" );
document.write( "You should get true results after simplifying everything.\r<BR>\n" );
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document.write( "--------------------------------------------------------\r<BR>\n" );
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document.write( "Answer: (x,y) = (-7, -6)<BR>\n" );
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