document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1068578>Question 1068578</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Two linear functions have the same rate of change and the same initial value. What do you know about the graph of the two lines? Explain.\r<BR>\n" );
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document.write( "Thanks! Please be detailed! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #683893 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=683893\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> if they have the same rate of change and the same initial value, then they are identical.\r<BR>\n" );
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document.write( "same rate of change means they have the same slope.\r<BR>\n" );
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document.write( "linear equations that have the same slope are either parallel or identical.\r<BR>\n" );
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document.write( "if they have a different y-intercept, they will be parallel, meaning that they will never intersect.\r<BR>\n" );
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document.write( "if they do intersect, and they have the same slope, then they have to be identical.\r<BR>\n" );
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document.write( "if they are identical, then they will intersect at all values of x.\r<BR>\n" );
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document.write( "if they intersect, they are either identical or not parallel.\r<BR>\n" );
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document.write( "if they are not parallel, they will have different slopes.\r<BR>\n" );
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document.write( "therefore, if they intersect and have the same slopes, they have to be identical, and they have to intersect at all values of x.\r<BR>\n" );
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document.write( "here's a graph of 2 lines that intersect but have different slopes.<BR>\n" );
document.write( "they intersect at the point (5,5) but have different slopes.<BR>\n" );
document.write( "they will only intersect in one point, that point being (5,5).\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2017/021601.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( "here's a graph of 2 lines that have the same slope but never intersect.<BR>\n" );
document.write( "they never intersect because they have the same slope but a different y-intercept.<BR>\n" );
document.write( "the first equation has a y-intercept of 0.<BR>\n" );
document.write( "the second equation has a y-intercept of 2.5<BR>\n" );
document.write( "since they are parallel, they can never intersect.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2017/021602.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( "here's the same 2 lines that have the same slope and intersect at the point (5,5).\r<BR>\n" );
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document.write( "they can only do that if they have the same slope and the same y-intercept.<BR>\n" );
document.write( "the two equations are identical.<BR>\n" );
document.write( "they are y = x<BR>\n" );
document.write( "the y-intercept is equal to 0 in both equations.+\r<BR>\n" );
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document.write( "this is shown in two graphs because the lines are identical.<BR>\n" );
document.write( "the first graph is the red equation.<BR>\n" );
document.write( "the second graph is the blue equation.<BR>\n" );
document.write( "they're the same equation because the lines are identical, meaning they have the same slope and the same y-intercept, which in this case is equal to 0.\r<BR>\n" );
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document.write( "each equation will draw the same exact line on the graph.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2017/021603.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2017/021604.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( " </SPAN>\n" );
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