document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=655658>Question 655658</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Write a linear function where f(2)=4,f(4)=2  and  f(3)=3. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #409227 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20850)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=409227\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%282%29=4 ALIGN=MIDDLE  ALT=\"f%282%29=4\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%284%29=2 ALIGN=MIDDLE  ALT=\"f%284%29=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%283%29=3 ALIGN=MIDDLE  ALT=\"f%283%29=3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then the graph of the function contains the point (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" >),(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" >), and (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" >). \r<BR>\n" );
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document.write( "Use the two point form of an equation of a straight line and the points (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" >),(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" >): \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29-y1=%28%28y1-y2%29%2F%28x1-x2%29%29%28x-x1%29 ALIGN=MIDDLE  ALT=\"f%28x%29-y1=%28%28y1-y2%29%2F%28x1-x2%29%29%28x-x1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > where (x1,y1)and (x2,y2) are the coordinates of the given points. \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29-4=%28%284-2%29%2F%282-4%29%29%28x-2%29 ALIGN=MIDDLE  ALT=\"f%28x%29-4=%28%284-2%29%2F%282-4%29%29%28x-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29-4=%282%2F-2%29%28x-2%29 ALIGN=MIDDLE  ALT=\"f%28x%29-4=%282%2F-2%29%28x-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29-4=-1%28x-2%29 ALIGN=MIDDLE  ALT=\"f%28x%29-4=-1%28x-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29-4=-x%2B2 ALIGN=MIDDLE  ALT=\"f%28x%29-4=-x%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=-x%2B2%2B4 ALIGN=MIDDLE  ALT=\"f%28x%29=-x%2B2%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28f%28x%29=-x%2B6%29 ALIGN=MIDDLE  ALT=\"highlight%28f%28x%29=-x%2B6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+600%2C+600%2C+-10%2C+10%2C+-10%2C+10%2C+-x%2B6%29+ ALIGN=MIDDLE  ALT=\"+graph%28+600%2C+600%2C+-10%2C+10%2C+-10%2C+10%2C+-x%2B6%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "here is a proof that all three of given points lie on this line\r<BR>\n" );
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document.write( "<TABLE  BORDER=1>\n" );
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document.write( " <TR><TD  ALIGN=MIDDLE>Solved by <A  rel=nofollow HREF=http://wiki.algebra.com/index.php/Tutoring_-_pluggable_solvers>pluggable</A> solver: <A  rel=nofollow HREF=To_determine_if_3_points_lie_in_a_line.solver>To determine if 3 points lie in a line</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT> The 3 points lie on a same plane. For all points to lie on a line they should satisfy the <A  rel=nofollow HREF=http://www.algebra.com/algebra/homework/Linear-equations/Polynomial-Expressions-and-Equations-in-Color.lesson>equation of a line</A>. Hence any two points taken on a line should calculate to the same slope of a line.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  In order to prove the 3 points to lie on a line, as there exists a unique line containing three points and every line has a unique <A  rel=nofollow HREF=http://www.algebra.com/algebra/homework/Graphs/slope.wikipedia>slope</A>. <BR>\n" );
document.write( "  Hence it will be sufficient to prove that the slope calculated taking 2 points at a time should be equal.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Slope of line taking points (X1,Y1) and (X2,Y2) is<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=slope+=+%28Y2-Y1%29%2F%28X2-X1%29 ALIGN=MIDDLE  ALT=\"slope+=+%28Y2-Y1%29%2F%28X2-X1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=slope+=+%28%282-4%29%2F%284-2%29%29+=+-1 ALIGN=MIDDLE  ALT=\"slope+=+%28%282-4%29%2F%284-2%29%29+=+-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >     ........................(1)<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Slope of line taking points (X3,Y3) and (X1,Y1) is<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=slope+=+%28Y3-Y1%29%2F%28X3-X1%29 ALIGN=MIDDLE  ALT=\"slope+=+%28Y3-Y1%29%2F%28X3-X1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=slope+=+%28%283-4%29%2F%283-2%29%29+=+-1 ALIGN=MIDDLE  ALT=\"slope+=+%28%283-4%29%2F%283-2%29%29+=+-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >    ........................(2)<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  From conditions (1) and (2)<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  The slopes are equal hence the 3 points can lie on same line.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  If the slope calculated from points (X2,Y2) and (X3,Y3) comes out to be same then it is confirmed that the 3 points lie on a same line.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=slope+=+%28Y3-Y2%29%2F%28X3-X2%29 ALIGN=MIDDLE  ALT=\"slope+=+%28Y3-Y2%29%2F%28X3-X2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=slope+=+%28%283-2%29%2F%283-4%29%29+=+-1 ALIGN=MIDDLE  ALT=\"slope+=+%28%283-2%29%2F%283-4%29%29+=+-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >    ........................(3)<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  From (1),(2) and (3)<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Hence, It is proved that the 3 points lie on same line.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  To read more on equations of a line refer to articles on <A  rel=nofollow HREF=http://www.algebra.com/algebra/homework/Graphs/Graphing-by-Slope-Intercept-Method.lesson>wikipedia</A><BR>\n" );
document.write( "  </TD></TR>\n" );
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document.write( " </SPAN>\n" );
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