document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1106903>Question 1106903</A>: <I><SPAN STYLE=\"word-break: break-all;\"> determine the value of k for which the system of linear equations has infinitely many solutions. then find all the solutions corresponding to this value of k<BR>\n" );
document.write( "3x+4y=12<BR>\n" );
document.write( "x+ky=4<BR>\n" );
document.write( " </SPAN>\n" );
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document.write( "3x+4y=12<BR>\n" );
document.write( "x+ky=4<br><BR>\n" );
document.write( "In order for the two equations to have infinitely many solutions, the equations must be equivalent.  Multiply the second equation by 3; now the two equations are<BR>\n" );
document.write( "3x+4y=12<BR>\n" );
document.write( "3x+3ky=12<br><BR>\n" );
document.write( "For the equations to be equivalent, we must have equal coefficients for y: 3k=4; so k = 4/3.<br><BR>\n" );
document.write( "I will guess that by \"listing all solutions\" you are looking for parametric equations.  To get them, solve the equation for y and then use x as the parameter:<br><BR>\n" );
document.write( "3x+4y = 12<BR>\n" );
document.write( "4y = -3x+12<BR>\n" );
document.write( "y = (-3/4)x+3<br><BR>\n" );
document.write( "The parametric equations representing all solutions are<BR>\n" );
document.write( "x = t<BR>\n" );
document.write( "y = (-3/4)t+3 </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );