document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1014327>Question 1014327</A>: <I><SPAN STYLE=\"word-break: break-all;\"> For what values of the coefficients a and b is the ordered pair (2,1) the solution to the linear system ax+by=-7 and 2ax-3by=1? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #630633 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=630633\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> your solution is (2,1).\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "this means x = 2 and y = 1\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "your equations are:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "ax + by = -7<BR>\n" );
document.write( "2ax - 3by = 1\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "since you know the value of x and y, you can replace x and y with their values in those equations.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "when x = 2 and y = 1, the equations of:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "ax + by = -7<BR>\n" );
document.write( "2ax - 3by = 1\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "become:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "2a + 1b = -7<BR>\n" );
document.write( "4a - 3b = 1\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the two equations you have to work with are now:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "2a + 1b = -7<BR>\n" );
document.write( "4a - 3b = 1\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "multiply both sides of the first equation by 2 and leave the second equation as is to get:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "4a + 2b = -14<BR>\n" );
document.write( "4a - 3b = 1\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "subtract the second equation from the first equation to get:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "5b = -15\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "solve for b to get b = -3\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "in the first original equation, solve for a to get:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "first original equation is 2a + 1b = -7<BR>\n" );
document.write( "when b = -3, this equation becomes 2a - 3 = -7<BR>\n" );
document.write( "add 3 to both sides of the equation to get 2a = -4<BR>\n" );
document.write( "solve for a to get a = -2\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "you have a = -2 and b = -3\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "go back to your original equations and replace a with -2 and b with -3.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "your original equations are:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "ax + by = -7<BR>\n" );
document.write( "2ax - 3by = 1\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "when a = -2 and b = -3, these equations become:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "-2x -3y = -7<BR>\n" );
document.write( "-4x + 9y = 1\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "to confirm that your values of a and b are correct, solve these two equations for x and y.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "you should get x = 2 and y = 1\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "start with:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "-2x - 3y = -7<BR>\n" );
document.write( "-4x + 9y = 1\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "multiply both sides of the first equation by 2 and leave the second equation as is to get:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "-4x - 6y = -14<BR>\n" );
document.write( "-4x + 9y = 1\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "subtract the second equation from the first to get:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "-15y = -15\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "solve for y to get y = 1.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "go back to your original first equation and solve for x.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "your first original equation in this series is -2x - 3y = -7<BR>\n" );
document.write( "when y = 1, this equation becomes -2x - 3 = -7<BR>\n" );
document.write( "add 3 to both sides of this equation to get -2x = -4<BR>\n" );
document.write( "divide both sides of this equation by -2 to get x = 2\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "you get x = 2 and y = 1 as you should, since that's the value of x and y that you started with when solving for a and b.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "your solution is:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "a = -2<BR>\n" );
document.write( "b = -3\r<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( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );