document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=150559>Question 150559</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I'm supposed to put the following formulas in y=mx+b form.  Then I'm supposed to plot the Y intercept.  Then to get the 2nd point I need to plot the rise and run of the two formulas.  Last I need to give the point at which the two lines intersect.  When I try these steps, it's not coming out right on my graph.\r<BR>\n" );
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document.write( "Plot the intersect points of the two equations<BR>\n" );
document.write( "2x + y = 4<BR>\n" );
document.write( "3x - 4y = 6<BR>\n" );
document.write( "Answer in book is (2,0)<BR>\n" );
document.write( "When I change the equations I get the following;<BR>\n" );
document.write( "y = -2x/1 -4<BR>\n" );
document.write( "y = 3/4x - 6/4 (I divided equation by -4 to get this solution)\r<BR>\n" );
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document.write( "Plot the intersect points of the two equations<BR>\n" );
document.write( "x + 3y = -6<BR>\n" );
document.write( "x - 3y = 0<BR>\n" );
document.write( "Answer in book is (-3,-1)<BR>\n" );
document.write( "When I change the equations I get the following;<BR>\n" );
document.write( "y = -1/3x - 2 (I divided equation by 3 to get this solution)<BR>\n" );
document.write( "y = 1/3x + 0 (I divided equation by -3 to get this solution)\r<BR>\n" );
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document.write( "Plot the intersect points of the two equations<BR>\n" );
document.write( "3x - 2y = -6<BR>\n" );
document.write( "-6x + 4y = 9<BR>\n" );
document.write( "Answer in book is No solution<BR>\n" );
document.write( "When I change the equations I get the following;<BR>\n" );
document.write( "y = 3/2x + 3 (I divided equation by -2 to get this solution)<BR>\n" );
document.write( "y = 3/2x + 9/4 (I divided equation by 4 to get this solution)\r<BR>\n" );
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document.write( "Plot the intersect points of the two equations<BR>\n" );
document.write( "x + 2y = 4<BR>\n" );
document.write( "-2x - 4y = -8<BR>\n" );
document.write( "Answer in book is Infinite<BR>\n" );
document.write( "When I change the equations I get the following;<BR>\n" );
document.write( "y = -1/2x + 2 (I divided equation by 2 to get this solution)<BR>\n" );
document.write( "y = 1/-2x + 2 (I divided equation by -4 to get this solution) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #110520 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=110520\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I'll do the first one to get you started\r<BR>\n" );
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document.write( "First let's graph the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x+%2B+y+=+4 ALIGN=MIDDLE  ALT=\"2x+%2B+y+=+4\"   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=2x+%2B+y+=+4 ALIGN=MIDDLE  ALT=\"2x+%2B+y+=+4\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the first equation.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=4-2x ALIGN=MIDDLE  ALT=\"y=4-2x\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x ALIGN=MIDDLE  ALT=\"2x\"   DISABLED_style= \"max-width:90%; height:auto;\" > from both sides.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-2x%2B4 ALIGN=MIDDLE  ALT=\"y=-2x%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" > Rearrange the terms.\r<BR>\n" );
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document.write( "Looking at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-2x%2B4 ALIGN=MIDDLE  ALT=\"y=-2x%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" > we can see that the equation is in slope-intercept form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=mx%2Bb ALIGN=MIDDLE  ALT=\"y=mx%2Bb\"   DISABLED_style= \"max-width:90%; height:auto;\" > where the slope is <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;\" > and the y-intercept is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=4 ALIGN=MIDDLE  ALT=\"b=4\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=4 ALIGN=MIDDLE  ALT=\"b=4\"   DISABLED_style= \"max-width:90%; height:auto;\" > this tells us that the y-intercept is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left(0,4\right)\">.Remember the y-intercept is the point where the graph intersects with the y-axis\r<BR>\n" );
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document.write( "So we have one point <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left(0,4\right)\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28500%2C500%2C-10%2C10%2C-10%2C10%2C%0D%0A++grid%281%29%2C%0D%0A++blue%28circle%280%2C4%2C.1%29%29%2C%0D%0A++blue%28circle%280%2C4%2C.12%29%29%2C%0D%0A++blue%28circle%280%2C4%2C.15%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now since the slope is comprised of the \"rise\" over the \"run\" this means\r<BR>\n" );
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document.write( "Also, because the slope is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2 ALIGN=MIDDLE  ALT=\"-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, this means:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=rise%2Frun=-2%2F1 ALIGN=MIDDLE  ALT=\"rise%2Frun=-2%2F1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "which shows us that the rise is -2 and the run is 1. This means that to go from point to point, we can go down 2  and over 1\r<BR>\n" );
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document.write( "So starting at <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left(0,4\right)\">, go down 2 units \r<BR>\n" );
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document.write( "and to the right 1 unit to get to the next point <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left(1,2\right)\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28500%2C500%2C-10%2C10%2C-10%2C10%2C%0D%0A++grid%281%29%2C%0D%0A++blue%28circle%280%2C4%2C.1%29%29%2C%0D%0A++blue%28circle%280%2C4%2C.12%29%29%2C%0D%0A++blue%28circle%280%2C4%2C.15%29%29%2C%0D%0A++blue%28circle%281%2C2%2C.15%2C1.5%29%29%2C%0D%0A++blue%28circle%281%2C2%2C.1%2C1.5%29%29%2C%0D%0A++blue%28arc%280%2C4%2B%28-2%2F2%29%2C2%2C-2%2C90%2C270%29%29%2C%0D%0A++blue%28arc%28%281%2F2%29%2C2%2C1%2C2%2C+0%2C180%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now draw a line through these points to graph <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-2x%2B4 ALIGN=MIDDLE  ALT=\"y=-2x%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=drawing%28500%2C500%2C-10%2C10%2C-10%2C10%2C%0D%0A++grid%281%29%2C%0D%0A++graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C-2x%2B4%29%2C%0D%0A++blue%28circle%280%2C4%2C.1%29%29%2C%0D%0A++blue%28circle%280%2C4%2C.12%29%29%2C%0D%0A++blue%28circle%280%2C4%2C.15%29%29%2C%0D%0A++blue%28circle%281%2C2%2C.15%2C1.5%29%29%2C%0D%0A++blue%28circle%281%2C2%2C.1%2C1.5%29%29%2C%0D%0A++blue%28arc%280%2C4%2B%28-2%2F2%29%2C2%2C-2%2C90%2C270%29%29%2C%0D%0A++blue%28arc%28%281%2F2%29%2C2%2C1%2C2%2C+0%2C180%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > So this is the graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-2x%2B4 ALIGN=MIDDLE  ALT=\"y=-2x%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" > through the points <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left(0,4\right)\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left(1,2\right)\">\r<BR>\n" );
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document.write( "Now let's graph the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x+-+4y+=+6 ALIGN=MIDDLE  ALT=\"3x+-+4y+=+6\"   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=3x+-+4y+=+6 ALIGN=MIDDLE  ALT=\"3x+-+4y+=+6\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the second equation.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4y=6-3x ALIGN=MIDDLE  ALT=\"-4y=6-3x\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x ALIGN=MIDDLE  ALT=\"3x\"   DISABLED_style= \"max-width:90%; height:auto;\" > from both sides.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%286-3x%29%2F%28-4%29 ALIGN=MIDDLE  ALT=\"y=%286-3x%29%2F%28-4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4 ALIGN=MIDDLE  ALT=\"-4\"   DISABLED_style= \"max-width:90%; height:auto;\" > to isolate <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   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=y=%286%29%2F%28-4%29-%283x%29%2F%28-4%29 ALIGN=MIDDLE  ALT=\"y=%286%29%2F%28-4%29-%283x%29%2F%28-4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Break up the fraction.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-3%2F2%2B%283x%29%2F%284%29 ALIGN=MIDDLE  ALT=\"y=-3%2F2%2B%283x%29%2F%284%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Reduce \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%283%2F4%29x-3%2F2 ALIGN=MIDDLE  ALT=\"y=%283%2F4%29x-3%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Rearrange the terms.\r<BR>\n" );
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document.write( "Looking at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%283%2F4%29x-3%2F2 ALIGN=MIDDLE  ALT=\"y=%283%2F4%29x-3%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > we can see that the equation is in slope-intercept form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=mx%2Bb ALIGN=MIDDLE  ALT=\"y=mx%2Bb\"   DISABLED_style= \"max-width:90%; height:auto;\" > where the slope is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=3%2F4 ALIGN=MIDDLE  ALT=\"m=3%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the y-intercept is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=-3%2F2 ALIGN=MIDDLE  ALT=\"b=-3%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=-3%2F2 ALIGN=MIDDLE  ALT=\"b=-3%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > this tells us that the y-intercept is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left(0,-\frac{3}{2}\right)\">.Remember the y-intercept is the point where the graph intersects with the y-axis\r<BR>\n" );
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document.write( "So we have one point <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left(0,-\frac{3}{2}\right)\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28500%2C500%2C-10%2C10%2C-10%2C10%2C%0D%0A++grid%281%29%2C%0D%0A++blue%28circle%280%2C-3%2F2%2C.1%29%29%2C%0D%0A++blue%28circle%280%2C-3%2F2%2C.12%29%29%2C%0D%0A++blue%28circle%280%2C-3%2F2%2C.15%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now since the slope is comprised of the \"rise\" over the \"run\" this means\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=slope=rise%2Frun ALIGN=MIDDLE  ALT=\"slope=rise%2Frun\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Also, because the slope is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2F4 ALIGN=MIDDLE  ALT=\"3%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" >, this means:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=rise%2Frun=3%2F4 ALIGN=MIDDLE  ALT=\"rise%2Frun=3%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "which shows us that the rise is 3 and the run is 4. This means that to go from point to point, we can go up 3  and over 4\r<BR>\n" );
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document.write( "So starting at <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left(0,-\frac{3}{2}\right)\">, go up 3 units \r<BR>\n" );
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document.write( "and to the right 4 units to get to the next point <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left(4,\frac{3}{2}\right)\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28500%2C500%2C-10%2C10%2C-10%2C10%2C%0D%0A++grid%281%29%2C%0D%0A++blue%28circle%280%2C-3%2F2%2C.1%29%29%2C%0D%0A++blue%28circle%280%2C-3%2F2%2C.12%29%29%2C%0D%0A++blue%28circle%280%2C-3%2F2%2C.15%29%29%2C%0D%0A++blue%28circle%284%2C3%2F2%2C.15%2C1.5%29%29%2C%0D%0A++blue%28circle%284%2C3%2F2%2C.1%2C1.5%29%29%2C%0D%0A++blue%28arc%280%2C-3%2F2%2B%283%2F2%29%2C2%2C3%2C90%2C270%29%29%2C%0D%0A++blue%28arc%28%284%2F2%29%2C3%2F2%2C4%2C2%2C+180%2C360%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now draw a line through these points to graph <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%283%2F4%29x-3%2F2 ALIGN=MIDDLE  ALT=\"y=%283%2F4%29x-3%2F2\"   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=drawing%28500%2C500%2C-10%2C10%2C-10%2C10%2C%0D%0A++grid%281%29%2C%0D%0A++graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C%283%2F4%29x-3%2F2%29%2C%0D%0A++blue%28circle%280%2C-3%2F2%2C.1%29%29%2C%0D%0A++blue%28circle%280%2C-3%2F2%2C.12%29%29%2C%0D%0A++blue%28circle%280%2C-3%2F2%2C.15%29%29%2C%0D%0A++blue%28circle%284%2C3%2F2%2C.15%2C1.5%29%29%2C%0D%0A++blue%28circle%284%2C3%2F2%2C.1%2C1.5%29%29%2C%0D%0A++blue%28arc%280%2C-3%2F2%2B%283%2F2%29%2C2%2C3%2C90%2C270%29%29%2C%0D%0A++blue%28arc%28%284%2F2%29%2C3%2F2%2C4%2C2%2C+180%2C360%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > So this is the graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%283%2F4%29x-3%2F2 ALIGN=MIDDLE  ALT=\"y=%283%2F4%29x-3%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > through the points <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left(0,-\frac{3}{2}\right)\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left(4,\frac{3}{2}\right)\">\r<BR>\n" );
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document.write( "======================================\r<BR>\n" );
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document.write( "So together the two graphs look like \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28500%2C500%2C-10%2C10%2C-10%2C10%2C%0D%0A++grid%281%29%2C%0D%0A++graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C-2x%2B4%2C%283%2F4%29x-3%2F2%29%2C%0D%0A++blue%28circle%282%2C0%2C.1%29%29%2C%0D%0A++blue%28circle%282%2C0%2C.12%29%29%2C%0D%0A++blue%28circle%282%2C0%2C.15%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-2x%2B4 ALIGN=MIDDLE  ALT=\"y=-2x%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" > (red) and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%283%2F4%29x-3%2F2 ALIGN=MIDDLE  ALT=\"y=%283%2F4%29x-3%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > (green)\r<BR>\n" );
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document.write( "From the graph, we can see that the two lines intersect at the point (2,0). So the solution of the system is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=2 ALIGN=MIDDLE  ALT=\"x=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=0 ALIGN=MIDDLE  ALT=\"y=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );