document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=256375>Question 256375</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If the graphs of the equations 2y + x + 3 = 0 and ax + 3y + 2 = 0 intersect at<BR>\n" );
document.write( "right angles in the plane, what is the numerical value of “a”? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #188496 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=188496\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> solve for the slope intercept form of the first equation of:\r<BR>\n" );
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document.write( "2y + x + 3 = 0. \r<BR>\n" );
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document.write( "subtract x and subtract 3 from both sides of the equation to get:\r<BR>\n" );
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document.write( "2y = -x - 3\r<BR>\n" );
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document.write( "divide both sides of the equation by 2 to get:\r<BR>\n" );
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document.write( "y = (-1/2)*x - (3/2)\r<BR>\n" );
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document.write( "solve for the slope intercept form of the second equation of:\r<BR>\n" );
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document.write( "ax + 3y + 2 = 0.\r<BR>\n" );
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document.write( "subtract ax and subtract + 2 from both sides of the equation to get:\r<BR>\n" );
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document.write( "3y = -ax - 2\r<BR>\n" );
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document.write( "divide both sides of the equation by 3 to get:\r<BR>\n" );
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document.write( "y = (-a/3)*x - (2/3)\r<BR>\n" );
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document.write( "slope intercept form of these equations is y = m*x + b where m is the slope and b is the y-intercept.\r<BR>\n" );
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document.write( "your 2 equations in slope intercept form are:\r<BR>\n" );
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document.write( "y = (-1/2)*x - (3/2)<BR>\n" );
document.write( "y = (-a/3)*x - (2/3)\r<BR>\n" );
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document.write( "if the graphs of these equations intersect at right angles, this means the lines formed by these equations are perpendicular to each other.\r<BR>\n" );
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document.write( "if the lines are perpendicular to each other, then their slopes are negative reciprocals of each other.\r<BR>\n" );
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document.write( "this means that the slope of one of the lines is equal to the negative reciprocal of the slope of the other line.\r<BR>\n" );
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document.write( "this means that:\r<BR>\n" );
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document.write( "(-1/2) = -1 / (-a/3)\r<BR>\n" );
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document.write( "multiply both sides of this equation by (-a/3) to get:\r<BR>\n" );
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document.write( "(-1/2)*(-a/3) = -1\r<BR>\n" );
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document.write( "multiply both sides of this equation by 6 to get:\r<BR>\n" );
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document.write( "(6)*(-1/2)*(-a/3) = -6\r<BR>\n" );
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document.write( "this simplifies to:\r<BR>\n" );
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document.write( "a = -6\r<BR>\n" );
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document.write( "the lines formed by these equations should be perpendicular when a = 6\r<BR>\n" );
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document.write( "take your second equation of:\r<BR>\n" );
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document.write( "y = (-a/3)*x - (2/3) and replace a with -6 to get:\r<BR>\n" );
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document.write( "y = (-(-6/3)*x) - (2/3) which becomes:\r<BR>\n" );
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document.write( "y = (6/3)*x - (2/3) which becomes:\r<BR>\n" );
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document.write( "y = 2*x - (2/3).\r<BR>\n" );
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document.write( "negative reciprocal of 2 is -(1/2) so the lines are perpendecular.\r<BR>\n" );
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document.write( "the two equation are:\r<BR>\n" );
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document.write( "y = (-1/2)*x - (3/2)<BR>\n" );
document.write( "y = (2)*x - (2/3)\r<BR>\n" );
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document.write( "this happens when a = -6.\r<BR>\n" );
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document.write( "graph of these 2 equations is shown below:\r<BR>\n" );
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document.write( " </SPAN>\n" );
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