document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=110225>Question 110225</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve the following system by using either addition or substitution. If a unique solution does not exist, state whether the system is dependent or inconsistent.\r<BR>\n" );
document.write( "\n" );
document.write( "10x + 2y = 7<BR>\n" );
document.write( "y = -5x + 3<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #80296 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20849)</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=80296\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10x+%2B+2y+=+7 ALIGN=MIDDLE  ALT=\"10x+%2B+2y+=+7\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+-5x+%2B+3 ALIGN=MIDDLE  ALT=\"y+=+-5x+%2B+3\"   DISABLED_style= \"max-width:90%; height:auto;\" >...........this equality we can write as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x+%2B+y+=+3 ALIGN=MIDDLE  ALT=\"5x+%2B+y+=+3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<TABLE  BORDER=1>\n" );
document.write( "\n" );
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><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=10%2Ax%2B2%2Ay=7 ALIGN=MIDDLE  ALT=\"10%2Ax%2B2%2Ay=7\"   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%2B1%2Ay=3 ALIGN=MIDDLE  ALT=\"5%2Ax%2B1%2Ay=3\"   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 10 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 10 and 5 is 10, we need to multiply both sides of the top equation by 1 and multiply both sides of the bottom equation by -2 like this:<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2A%2810%2Ax%2B2%2Ay%29=%287%29%2A1 ALIGN=MIDDLE  ALT=\"1%2A%2810%2Ax%2B2%2Ay%29=%287%29%2A1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply the top equation (both sides) by 1<BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2%2A%285%2Ax%2B1%2Ay%29=%283%29%2A-2 ALIGN=MIDDLE  ALT=\"-2%2A%285%2Ax%2B1%2Ay%29=%283%29%2A-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply the bottom equation (both sides) by -2<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=10%2Ax%2B2%2Ay=7 ALIGN=MIDDLE  ALT=\"10%2Ax%2B2%2Ay=7\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-10%2Ax-2%2Ay=-6 ALIGN=MIDDLE  ALT=\"-10%2Ax-2%2Ay=-6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Notice how 10 and -10 and 7 and -2 add to zero (ie <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10%2B-10=0 ALIGN=MIDDLE  ALT=\"10%2B-10=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2B-2=0 ALIGN=MIDDLE  ALT=\"2%2B-2=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  However 7 and -6 add to 1 (ie <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7%2B-6=1 ALIGN=MIDDLE  ALT=\"7%2B-6=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >);<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  So we're left with <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0=1 ALIGN=MIDDLE  ALT=\"0=1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  which means no value of x or y value will satisfy the system of equations. So there are no solutions<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  So this system is <a  rel=nofollow HREF=http://www.algebra.com/algebra/homework/coordinate/lessons/Types-of-systems-inconsistent-dependent-independent.lesson>inconsistent</a></TD></TR>\n" );
document.write( "</TABLE><BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );