document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=622447>Question 622447</A>: <I><SPAN STYLE=\"word-break: break-all;\"> y=4x+4<BR>\n" );
document.write( "3x+2y=12\r<BR>\n" );
document.write( "\n" );
document.write( "solve this as a linear equation and graph. don't know how to answer this. please help. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #391684 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=391684\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=4x%2B4 ALIGN=MIDDLE  ALT=\"y=4x%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a linear equation.<BR>\n" );
document.write( "So is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%2B2y=12+ ALIGN=MIDDLE  ALT=\"3x%2B2y=12+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Linear equations involve one or more variables (usually x, and/or y, and/or z) with no exponents.<BR>\n" );
document.write( "A linear equation involving only one or two variables can be graphed as a straight line. (If a linear equation has three or more variables, you cannot graph it).<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28y=4x%2B4%2C3x%2B2y=12%29 ALIGN=MIDDLE  ALT=\"system%28y=4x%2B4%2C3x%2B2y=12%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a system of linear equations with two variables.<BR>\n" );
document.write( "There are several ways to solve such a system of linear equations. Sometimes you can use graphing to get to the solution. Even when graphing does not lead you to the solution, a graph may help to figure out the problem.<BR>\n" );
document.write( "Each equation can be graphed as a line.<BR>\n" );
document.write( "If the lines are parallel, there is no solution.<BR>\n" );
document.write( "If the same line is the graph for both equations, all the points in that line represent the infinite solutions to the system of equations.<BR>\n" );
document.write( "If the lines intersect, the intersection point represents the unique solution to the system. In that case, if the values for x and y that correspond to the intersection can be clearly read from the graph, you have a tentative solution. It still needs to be verified by substituting into the equations.<BR>\n" );
document.write( " <BR>\n" );
document.write( "GRAPHING THE SYSTEM:To graph each line, you need only two points.<BR>\n" );
document.write( "You can chose any points, but you may want to chose points that make calculations easy and give you a convenient graph. (You can often get good points by making x=0 and/or y=0).<BR>\n" );
document.write( "For equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%2B2y=12+ ALIGN=MIDDLE  ALT=\"3x%2B2y=12+\"   DISABLED_style= \"max-width:90%; height:auto;\" >:<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;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2y=12 ALIGN=MIDDLE  ALT=\"2y=12\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=6 ALIGN=MIDDLE  ALT=\"y=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > gives you point (0,6).<BR>\n" );
document.write( "<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;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x=13 ALIGN=MIDDLE  ALT=\"3x=13\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=4 ALIGN=MIDDLE  ALT=\"x=4\"   DISABLED_style= \"max-width:90%; height:auto;\" > gives you point (4,0).<BR>\n" );
document.write( "The graph for that line (with circles around the points used) is:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-2%2C8%2C-1%2C9%2C%0D%0Agrid%281%29%2C%0D%0Ablue%28circle%280%2C6%2C0.2%29%29%2Cblue%28circle%284%2C0%2C0.2%29%29%2C%0D%0Ablue%28line%28-2%2C9%2C8%2C-6%29%29+%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "For equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=4x%2B4 ALIGN=MIDDLE  ALT=\"y=4x%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" >:<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;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=4 ALIGN=MIDDLE  ALT=\"y=4\"   DISABLED_style= \"max-width:90%; height:auto;\" > gives you point (0,4).<BR>\n" );
document.write( "<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;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%2B4=0 ALIGN=MIDDLE  ALT=\"4x%2B4=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x=-4 ALIGN=MIDDLE  ALT=\"4x=-4\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-1 ALIGN=MIDDLE  ALT=\"x=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > gives you point (-1,0).<BR>\n" );
document.write( "The graph for the whole system, with the new line graphed in green, looks like this:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-2%2C8%2C-1%2C9%2C%0D%0Agrid%281%29%2C%0D%0Ablue%28circle%280%2C6%2C0.2%29%29%2Cblue%28circle%284%2C0%2C0.2%29%29%2C%0D%0Ablue%28line%28-2%2C9%2C8%2C-6%29%29%2C%0D%0Agreen%28circle%280%2C4%2C0.2%29%29%2Cgreen%28circle%28-1%2C0%2C0.2%29%29%2C%0D%0Agreen%28line%28-2%2C-4%2C2%2C12%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > We see that there is one point that represents the solution, but can only make a wild guess about the coordinates of that point.<BR>\n" );
document.write( " <BR>\n" );
document.write( "SOLVING BY SUBSTITUTION:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=4x%2B4 ALIGN=MIDDLE  ALT=\"y=4x%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" > gives you an expression for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" > that can be substituted into<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%2B2y=12+ ALIGN=MIDDLE  ALT=\"3x%2B2y=12+\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%2B2%284x%2B4%29=12 ALIGN=MIDDLE  ALT=\"3x%2B2%284x%2B4%29=12\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%2B8x%2B8=12 ALIGN=MIDDLE  ALT=\"3x%2B8x%2B8=12\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=11x%2B8=12 ALIGN=MIDDLE  ALT=\"11x%2B8=12\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=11x=12-8 ALIGN=MIDDLE  ALT=\"11x=12-8\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=11x=4 ALIGN=MIDDLE  ALT=\"11x=4\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=11x%2F11=4%2F11 ALIGN=MIDDLE  ALT=\"11x%2F11=4%2F11\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28x=4%2F11%29 ALIGN=MIDDLE  ALT=\"highlight%28x=4%2F11%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now we substitute that value into <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=4x%2B4 ALIGN=MIDDLE  ALT=\"y=4x%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" >  to get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=4%284%2F11%29%2B4 ALIGN=MIDDLE  ALT=\"y=4%284%2F11%29%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=16%2F11%2B4 ALIGN=MIDDLE  ALT=\"y=16%2F11%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28y=60%2F11%29 ALIGN=MIDDLE  ALT=\"highlight%28y=60%2F11%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28y=5%265%2F11%29 ALIGN=MIDDLE  ALT=\"highlight%28y=5%265%2F11%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So the intersection of those two lines is the point with<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=4%2F11 ALIGN=MIDDLE  ALT=\"x=4%2F11\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=60%2F11=5%265%2F11 ALIGN=MIDDLE  ALT=\"y=60%2F11=5%265%2F11\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "We could never have figured that out from the graph. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );