document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1204488>Question 1204488</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve the following system of equations graphically on the set of axes below.\r<BR>\n" );
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document.write( "y=x+8<BR>\n" );
document.write( "y=−2x−4 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #840783 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=840783\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "If your teacher allows technology to graph, there are many options to pick from. My top two favorites are Desmos and GeoGebra. <BR>\n" );
document.write( "If you are in an exam setting, then use of a TI83 or TI84 or similar is probably the better route.\r<BR>\n" );
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document.write( "Here's the Desmos link to the interactive graph<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://www.desmos.com/calculator/wcygja34rd\">https://www.desmos.com/calculator/wcygja34rd</a>\r<BR>\n" );
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document.write( "While using tech is fine for a quick solution, it doesn't help show how the graphing process works. <BR>\n" );
document.write( "The next section will cover this.\r<BR>\n" );
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document.write( "Each equation given to us is linear because they fit the slope intercept form y = mx+b<BR>\n" );
document.write( "All we need at minimum are two points to plot a line.\r<BR>\n" );
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document.write( "Each point is a pair of coordinates (x,y).<BR>\n" );
document.write( "To generate any given point on the line, pick x values at random to plug into the equation to find its paired y value.\r<BR>\n" );
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document.write( "For example, let's go for x = 0<BR>\n" );
document.write( "y = x+8<BR>\n" );
document.write( "y = 0+8<BR>\n" );
document.write( "y = 8<BR>\n" );
document.write( "The input x = 0, leads to y = 8<BR>\n" );
document.write( "We form the ordered pair (x,y) = (0,8)\r<BR>\n" );
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document.write( "Repeat the process with x = 1<BR>\n" );
document.write( "y = x+8<BR>\n" );
document.write( "y = 1+8<BR>\n" );
document.write( "y = 9<BR>\n" );
document.write( "The point (1,9) is also on the line y = x+8\r<BR>\n" );
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document.write( "Plot the two points (0,8) and (1,9)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%0D%0Adrawing%28400%2C400%2C-5%2C5%2C-2%2C11%2C%0D%0Agraph%28400%2C400%2C-5%2C5%2C-2%2C11%29%2C%0D%0Acircle%280%2C8%2C0.05%29%2Ccircle%280%2C8%2C0.07%29%2Ccircle%280%2C8%2C0.09%29%2Ccircle%280%2C8%2C0.11%29%2Ccircle%281%2C9%2C0.05%29%2Ccircle%281%2C9%2C0.07%29%2Ccircle%281%2C9%2C0.09%29%2Ccircle%281%2C9%2C0.11%29%0D%0A%29%0D%0A%0D%0A ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Then draw a straight line through those points to finish up the graph of y = x+8<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%0D%0Adrawing%28400%2C400%2C-5%2C5%2C-2%2C11%2C%0D%0Agraph%28400%2C400%2C-5%2C5%2C-2%2C11%2C-1000%2Cx%2B8%29%2C%0D%0Acircle%280%2C8%2C0.05%29%2Ccircle%280%2C8%2C0.07%29%2Ccircle%280%2C8%2C0.09%29%2Ccircle%280%2C8%2C0.11%29%2Ccircle%281%2C9%2C0.05%29%2Ccircle%281%2C9%2C0.07%29%2Ccircle%281%2C9%2C0.09%29%2Ccircle%281%2C9%2C0.11%29%0D%0A%29%0D%0A%0D%0A ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "If you follow the same outline for the other equation y = -2x-4, then you should find the two points (0,-4) and (1,-6)<BR>\n" );
document.write( "Let me know if you need to see the steps.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%0D%0Adrawing%28400%2C400%2C-5%2C5%2C-11%2C2%2C%0D%0Agraph%28400%2C400%2C-5%2C5%2C-11%2C2%2C-1000%2C-1000%2C-2x-4%29%2C%0D%0Acircle%280%2C-4%2C0.05%29%2Ccircle%280%2C-4%2C0.07%29%2Ccircle%280%2C-4%2C0.09%29%2Ccircle%280%2C-4%2C0.11%29%2Ccircle%281%2C-6%2C0.05%29%2Ccircle%281%2C-6%2C0.07%29%2Ccircle%281%2C-6%2C0.09%29%2Ccircle%281%2C-6%2C0.11%29%0D%0A%29%0D%0A%0D%0A ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Here's what it looks like when graphing both equations on the same xy grid.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28400%2C400%2C-5%2C5%2C-7%2C11%2C%0D%0Agraph%28400%2C400%2C-5%2C5%2C-7%2C11%2C-1000%2Cx%2B8%2C-2x-4%29%2C%0D%0Acircle%28-4%2C4%2C0.05%29%2Ccircle%28-4%2C4%2C0.07%29%2Ccircle%28-4%2C4%2C0.09%29%2Ccircle%28-4%2C4%2C0.11%29%2C%0D%0Alocate%28-4%2B0.3%2C4%2C%22%28-4%2C4%29%22%29%0D%0A%29%0D%0A ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "y = x+8 in green<BR>\n" );
document.write( "y = -2x-4 in blue<BR>\n" );
document.write( "The intersection point (-4,4) is the solution of the system.<BR>\n" );
document.write( "It means x = -4 and y = 4 pair up together to make both equations true.\r<BR>\n" );
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document.write( "Let's check that claim algebraically.<BR>\n" );
document.write( "y = x+8<BR>\n" );
document.write( "4 = -4+8<BR>\n" );
document.write( "4 = 4<BR>\n" );
document.write( "The first equation has been confirmed.<BR>\n" );
document.write( "Now onto the second equation.<BR>\n" );
document.write( "y = -2x-4<BR>\n" );
document.write( "4 = -2*(-4)-4<BR>\n" );
document.write( "4 = 8-4<BR>\n" );
document.write( "4 = 4<BR>\n" );
document.write( "This has been confirmed as well.<BR>\n" );
document.write( "Both equations have been confirmed true.<BR>\n" );
document.write( "Therefore, we have confirmed the solution (-4, 4).<BR>\n" );
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