document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=203229>Question 203229</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The points (1, 3), (2, 5), (8, 17), (50, 101), and (1000, 2001) all lie on line M.<BR>\n" );
document.write( "The points (1, -1), (2, -3), (8, -15), (50, -99), and (1000, -1999) all lie on line N.<BR>\n" );
document.write( "a.	Form the equations of both the lines.  Show your work.  <BR>\n" );
document.write( "b.	What are the co-ordinates of the point of intersection of lines M and N?<BR>\n" );
document.write( "c.	Write the co-ordinates of the intersections of lines M and N with the x-axis.<BR>\n" );
document.write( "d.	Write the co-ordinates of the intersection of lines M and N with the y-axis. Could you help me with these? This is very difficult for me. You are very much appreciated.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #153335 by </B><A HREF=/tutors/aboutme.mpl?userid=Earlsdon><B>Earlsdon(6294)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Earlsdon><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=153335\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Since we are talking about linear equations, we need only two points to define the line and we can write the equation of the line if we know its slope and its y-intercept.<BR>\n" );
document.write( "We can write the equations in the slope-intercept form: y = mx+b where m = slope and b = y-intercept.<BR>\n" );
document.write( "For line M, pick any two of the given five points to find the slope (m) of the line:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m+=+%28y%5B2%5D-y%5B1%5D%29%2F%28x%5B2%5D-x%5B1%5D%29 ALIGN=MIDDLE  ALT=\"m+=+%28y%5B2%5D-y%5B1%5D%29%2F%28x%5B2%5D-x%5B1%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Let's pick the first two points from the list (1, 3) and (2, 5) and substitute their x- and y-cordinates into the slope formula:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m+=+%285-3%29%2F%282-1%29 ALIGN=MIDDLE  ALT=\"m+=+%285-3%29%2F%282-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m+=+2%2F1 ALIGN=MIDDLE  ALT=\"m+=+2%2F1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m+=+2 ALIGN=MIDDLE  ALT=\"m+=+2\"   DISABLED_style= \"max-width:90%; height:auto;\" > so, we have, so far...<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+2x%2Bb ALIGN=MIDDLE  ALT=\"y+=+2x%2Bb\"   DISABLED_style= \"max-width:90%; height:auto;\" > To find b, the y-intercept, you can use the x- and y-coordinates of either one of the two points (or any of the other given points for that matter, as long as they lie on the line).  Let's use the first point in the list (1, 3):<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3+=+2%281%29%2Bb ALIGN=MIDDLE  ALT=\"3+=+2%281%29%2Bb\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract 2 from both sides of the equation.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1+=+b ALIGN=MIDDLE  ALT=\"1+=+b\"   DISABLED_style= \"max-width:90%; height:auto;\" > so the final equation for line M is:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28y+=+2x%2B1%29 ALIGN=MIDDLE  ALT=\"highlight%28y+=+2x%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Similarly, for line N:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m+=+%28y%5B2%5D-y%5B1%5D%29%2F%28x%5B2%5D-x%5B1%5D%29 ALIGN=MIDDLE  ALT=\"m+=+%28y%5B2%5D-y%5B1%5D%29%2F%28x%5B2%5D-x%5B1%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Pick points (1, -1) and (2, -3), substitute their x- and y-coordinates:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m+=+%28-3-%28-1%29%29%2F%282-1%29 ALIGN=MIDDLE  ALT=\"m+=+%28-3-%28-1%29%29%2F%282-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m+=+%28-3%2B1%29%2F1%29 ALIGN=MIDDLE  ALT=\"m+=+%28-3%2B1%29%2F1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m+=+-2 ALIGN=MIDDLE  ALT=\"m+=+-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > so...<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+-2x%2Bb ALIGN=MIDDLE  ALT=\"y+=+-2x%2Bb\"   DISABLED_style= \"max-width:90%; height:auto;\" > use the second point (2, -3) to find b, substitute:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3+=+-2%282%29%2Bb ALIGN=MIDDLE  ALT=\"-3+=+-2%282%29%2Bb\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3+=+-4%2Bb ALIGN=MIDDLE  ALT=\"-3+=+-4%2Bb\"   DISABLED_style= \"max-width:90%; height:auto;\" > Add 4 to both sides.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1+=+b ALIGN=MIDDLE  ALT=\"1+=+b\"   DISABLED_style= \"max-width:90%; height:auto;\" > so the final equation for line N is:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight_green%28y+=+-2x%2B1%29 ALIGN=MIDDLE  ALT=\"highlight_green%28y+=+-2x%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "-------------------------------------<BR>\n" );
document.write( "The point of intersection of the two lines is found by solving the system of linear equations just developed.  This is esily done by setting the two equations equal and solving for x.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%2B1+=+-2x%2B1 ALIGN=MIDDLE  ALT=\"2x%2B1+=+-2x%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Add 2x to both sides.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%2B1+=+1 ALIGN=MIDDLE  ALT=\"4x%2B1+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract 1 from both sides.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x+=+0 ALIGN=MIDDLE  ALT=\"4x+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >so...<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+0 ALIGN=MIDDLE  ALT=\"x+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > now substitute this into either one of the two equations and solve for y:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+2%280%29%2B1 ALIGN=MIDDLE  ALT=\"y+=+2%280%29%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+1 ALIGN=MIDDLE  ALT=\"y+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The point of intersection of the two lines is (0, 1)<BR>\n" );
document.write( "--------------------------------------------<BR>\n" );
document.write( "The line intersect the x-axis at y = 0, so, set y = 0 in each equation and solve for x.<BR>\n" );
document.write( "For line M:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+2x%2B1 ALIGN=MIDDLE  ALT=\"y+=+2x%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Substitute y = o.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+=+2x%2B1 ALIGN=MIDDLE  ALT=\"0+=+2x%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract 1 from both sides.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1+=+2x ALIGN=MIDDLE  ALT=\"-1+=+2x\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by 2.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2F2+=+x ALIGN=MIDDLE  ALT=\"-1%2F2+=+x\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28x+=+-1%2F2%29 ALIGN=MIDDLE  ALT=\"highlight%28x+=+-1%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "For line N:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+-2x%2B1 ALIGN=MIDDLE  ALT=\"y+=+-2x%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Substitute y = 0.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+=+-2x%2B1 ALIGN=MIDDLE  ALT=\"0+=+-2x%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract 1 from both sides.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1+=+-2x ALIGN=MIDDLE  ALT=\"-1+=+-2x\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by -2.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F2+=+x ALIGN=MIDDLE  ALT=\"1%2F2+=+x\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight_green%28x+=+1%2F2%29 ALIGN=MIDDLE  ALT=\"highlight_green%28x+=+1%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "---------------------------------------<BR>\n" );
document.write( "Remember that the intersection of the lines with the y-axis is called the y-intercept or b in the equation of the lines.<BR>\n" );
document.write( "Foe line M: b = 1.<BR>\n" );
document.write( "For line N: b = 1.<BR>\n" );
document.write( "----------------------------------<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28400%2C400%2C-5%2C5%2C-5%2C5%2C2x%2B1%2C-2x%2B1%29 ALIGN=MIDDLE  ALT=\"graph%28400%2C400%2C-5%2C5%2C-5%2C5%2C2x%2B1%2C-2x%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );