document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=110680>Question 110680</A>: <I><SPAN STYLE=\"word-break: break-all;\"> y=1/3x + 2/3<BR>\n" );
document.write( "y=5/7x-2 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #80656 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20850)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=80656\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=1%2F3x+%2B+2%2F3 ALIGN=MIDDLE  ALT=\"y=1%2F3x+%2B+2%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=5%2F7x%962 ALIGN=MIDDLE  ALT=\"y=5%2F7x%962\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "In standard form:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-%281%2F3%29x+%2B+y=+2%2F3 ALIGN=MIDDLE  ALT=\"-%281%2F3%29x+%2B+y=+2%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-%285%2F7%29x+%2B+y=+-2 ALIGN=MIDDLE  ALT=\"-%285%2F7%29x+%2B+y=+-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Solution:\r<BR>\n" );
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document.write( " <TR><TD  ALIGN=MIDDLE>Solved by <A  rel=nofollow HREF=http://wiki.algebra.com/index.php/Tutoring_-_pluggable_solvers>pluggable</A> solver: <A  rel=nofollow HREF=solving_linear_system_by_elimination.solver>Solving a System of Linear Equations by Elimination/Addition</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT>TEST<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28-1%2F3%29%29%2Ax%2B%281%29%2Ay=%282%2F3%29 ALIGN=MIDDLE  ALT=\"%28%28-1%2F3%29%29%2Ax%2B%281%29%2Ay=%282%2F3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the first equation<BR>\n" );
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document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%28%28%28-1%2F3%29%29%2Ax%2B%281%29%2Ay%29=%283%29%2A%28%282%2F3%29%29 ALIGN=MIDDLE  ALT=\"3%28%28%28-1%2F3%29%29%2Ax%2B%281%29%2Ay%29=%283%29%2A%28%282%2F3%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply both sides by the LCD 3<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2Ax%2B3%2Ay=2 ALIGN=MIDDLE  ALT=\"-1%2Ax%2B3%2Ay=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >Distribute and simplify<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  ------------------------------------------<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28-5%2F7%29%29%2Ax%2B%281%29%2Ay=%28-2%29 ALIGN=MIDDLE  ALT=\"%28%28-5%2F7%29%29%2Ax%2B%281%29%2Ay=%28-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the second equation<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7%28%28%28-5%2F7%29%29%2Ax%2B%281%29%2Ay%29=%287%29%2A%28%28-2%29%29 ALIGN=MIDDLE  ALT=\"7%28%28%28-5%2F7%29%29%2Ax%2B%281%29%2Ay%29=%287%29%2A%28%28-2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply both sides by the LCD 7<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-5%2Ax%2B7%2Ay=0 ALIGN=MIDDLE  ALT=\"-5%2Ax%2B7%2Ay=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Distribute and simplify<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
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document.write( "  -----------------------------------------<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Lets start with the given system of linear equations<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2Ax%2B3%2Ay=2 ALIGN=MIDDLE  ALT=\"-1%2Ax%2B3%2Ay=2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-5%2Ax%2B7%2Ay=0 ALIGN=MIDDLE  ALT=\"-5%2Ax%2B7%2Ay=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  In order to solve for one variable, we must eliminate the other variable. So if we wanted to solve for y, we would have to eliminate x (or vice versa).<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  So lets eliminate x. In order to do that, we need to have both x coefficients that are equal but have opposite signs (for instance 2 and -2 are equal but have opposite signs). This way they will add to zero.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  So to make the x coefficients equal but opposite, we need to multiply both x coefficients by some number to get them to an equal number. So if we wanted to get -1 and -5 to some equal number, we could try to get them to the LCM.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Since the LCM of -1 and -5 is 5, we need to multiply both sides of the top equation by -5 and multiply both sides of the bottom equation by 1 like this:<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-5%2A%28-1%2Ax%2B3%2Ay%29=%282%29%2A-5 ALIGN=MIDDLE  ALT=\"-5%2A%28-1%2Ax%2B3%2Ay%29=%282%29%2A-5\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply the top equation (both sides) by -5<BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2A%28-5%2Ax%2B7%2Ay%29=%280%29%2A1 ALIGN=MIDDLE  ALT=\"1%2A%28-5%2Ax%2B7%2Ay%29=%280%29%2A1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply the bottom equation (both sides) by 1<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  So after multiplying we get this:<BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2Ax-15%2Ay=-10 ALIGN=MIDDLE  ALT=\"5%2Ax-15%2Ay=-10\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-5%2Ax%2B7%2Ay=0 ALIGN=MIDDLE  ALT=\"-5%2Ax%2B7%2Ay=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Notice how 5 and -5 add to zero (ie <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2B-5=0 ALIGN=MIDDLE  ALT=\"5%2B-5=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Now add the equations together. In order to add 2 equations, group like terms and combine them<BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%285%2Ax-5%2Ax%29-15%2Ay%2B7%2Ay%29=-10%2B0 ALIGN=MIDDLE  ALT=\"%285%2Ax-5%2Ax%29-15%2Ay%2B7%2Ay%29=-10%2B0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%285-5%29%2Ax-15%2B7%29y=-10%2B0 ALIGN=MIDDLE  ALT=\"%285-5%29%2Ax-15%2B7%29y=-10%2B0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cross%285%2B-5%29%2Ax%2B%28-15%2B7%29%2Ay=-10%2B0 ALIGN=MIDDLE  ALT=\"cross%285%2B-5%29%2Ax%2B%28-15%2B7%29%2Ay=-10%2B0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Notice the x coefficients add to zero and cancel out. This means we've eliminated x altogether.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  So after adding and canceling out the x terms we're left with:<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-8%2Ay=-10 ALIGN=MIDDLE  ALT=\"-8%2Ay=-10\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-10%2F-8 ALIGN=MIDDLE  ALT=\"y=-10%2F-8\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-8 ALIGN=MIDDLE  ALT=\"-8\"   DISABLED_style= \"max-width:90%; height:auto;\" > to solve for y<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=5%2F4 ALIGN=MIDDLE  ALT=\"y=5%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > Reduce<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Now plug this answer into the top equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2Ax%2B3%2Ay=2 ALIGN=MIDDLE  ALT=\"-1%2Ax%2B3%2Ay=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > to solve for x<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2Ax%2B3%285%2F4%29=2 ALIGN=MIDDLE  ALT=\"-1%2Ax%2B3%285%2F4%29=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=5%2F4 ALIGN=MIDDLE  ALT=\"y=5%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2Ax%2B15%2F4=2 ALIGN=MIDDLE  ALT=\"-1%2Ax%2B15%2F4=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2Ax%2B15%2F4=2 ALIGN=MIDDLE  ALT=\"-1%2Ax%2B15%2F4=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Reduce<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2Ax=2-15%2F4 ALIGN=MIDDLE  ALT=\"-1%2Ax=2-15%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15%2F4 ALIGN=MIDDLE  ALT=\"15%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > from both sides<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2Ax=8%2F4-15%2F4 ALIGN=MIDDLE  ALT=\"-1%2Ax=8%2F4-15%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > Make 2 into a fraction with a denominator of 4<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2Ax=-7%2F4 ALIGN=MIDDLE  ALT=\"-1%2Ax=-7%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine the terms on the right side<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cross%28%281%2F-1%29%28-1%29%29%2Ax=%28-7%2F4%29%281%2F-1%29 ALIGN=MIDDLE  ALT=\"cross%28%281%2F-1%29%28-1%29%29%2Ax=%28-7%2F4%29%281%2F-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply both sides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F-1 ALIGN=MIDDLE  ALT=\"1%2F-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >. This will cancel out <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1 ALIGN=MIDDLE  ALT=\"-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > on the left side.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=7%2F4 ALIGN=MIDDLE  ALT=\"x=7%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply the terms on the right side<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  So our answer is <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=7%2F4 ALIGN=MIDDLE  ALT=\"x=7%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" >,  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=5%2F4 ALIGN=MIDDLE  ALT=\"y=5%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  which also looks like<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7%2F4 ALIGN=MIDDLE  ALT=\"7%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2F4 ALIGN=MIDDLE  ALT=\"5%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Notice if we graph the equations (if you need help with graphing, check out this <a  rel=nofollow HREF=http://www.algebra.com/algebra/homework/Linear-equations/graphing-linear-equations.solver>solver</a>)<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2Ax%2B3%2Ay=2 ALIGN=MIDDLE  ALT=\"-1%2Ax%2B3%2Ay=2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-5%2Ax%2B7%2Ay=0 ALIGN=MIDDLE  ALT=\"-5%2Ax%2B7%2Ay=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  we get <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%0D%0A++drawing%28+500%2C+600%2C+-11.75%2C+11.75%2C+-11.25%2C+11.25%2C%0D%0A++++graph%28+500%2C+600%2C+-11.75%2C+11.75%2C+-11.25%2C+11.25%2C+%282--1%2Ax%29%2F3%2C+%280--5%2Ax%29%2F7+%29%2C%0D%0A++++blue%28++circle%28+1.75%2C+1.25%2C+0.156666666666667+%29+%29+%0D%0A++%29%0D%0A++ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2Ax%2B3%2Ay=2 ALIGN=MIDDLE  ALT=\"-1%2Ax%2B3%2Ay=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > (red) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-5%2Ax%2B7%2Ay=0 ALIGN=MIDDLE  ALT=\"-5%2Ax%2B7%2Ay=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (green) (hint: you may have to solve for y to graph these) and the intersection of the lines (blue circle).<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  and we can see that the two equations intersect at (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7%2F4 ALIGN=MIDDLE  ALT=\"7%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2F4 ALIGN=MIDDLE  ALT=\"5%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" >). This verifies our answer.</TD></TR>\n" );
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