document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1007035>Question 1007035</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Show and explain how replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions as the one shown.<BR>\n" );
document.write( "8x + 7y = 39<BR>\n" );
document.write( "4x – 14y = –68 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #623070 by </B><A HREF=/tutors/aboutme.mpl?userid=richard1234><B>richard1234(7193)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=richard1234><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=richard1234><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=623070\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Assume that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large (x_0, y_0)\"> is a solution to the system\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large a_1 x + a_2 y = c_1\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large b_1 x + b_2 y = c_2\">\r<BR>\n" );
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document.write( "Then all that needs to be done is show that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large (x_0, y_0)\"> is a solution to the system\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large a_1 x + a_2 y = c_1\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large (a_1 + kb_1)x + (a_2 + kb_2)y = c_1 + kc_2\"> (for some constant k)\r<BR>\n" );
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document.write( "Note that the second equation was formed by taking the first and adding k times the second. However this pretty easily holds true from linearity as the second equation is equivalent to\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large (a_1x + a_2y) + k(b_1x + b_2y) = c_1 + kc_2\">\r<BR>\n" );
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document.write( "which is true. </SPAN>\n" );
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