document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=229937>Question 229937</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Given the following system of linear equation, make comment about the lines and the number of solutions this system has.\r<BR>\n" );
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document.write( "y = -3x - 12<BR>\n" );
document.write( "y = 2x + 12  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #170361 by </B><A HREF=/tutors/aboutme.mpl?userid=drj><B>drj(1380)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=drj><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=drj><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=170361\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Given the following system of linear equation, make comment about the lines and the number of solutions this system has.\r<BR>\n" );
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document.write( "y = -3x - 12  Equation A<BR>\n" );
document.write( "y = 2x + 12   Equation B\r<BR>\n" );
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document.write( "Step 1.  The above lines in given in slope intercept form given as y=mx+b where m is the slope and b is the y-intercept when x=0 or at point (0,b).\r<BR>\n" );
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document.write( "Step 2.  The slope of Equation A is m1=-3 and the slope of Equation B is m2=2.  So they are not parallel since the parallel lines have the same slope.  The lines are not perpendicular since the product of their slopes is not equal to -1 since m1*m2 does not equal -1.\r<BR>\n" );
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document.write( "Step 3.  The lines intersect at a point so there is one solution.\r<BR>\n" );
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document.write( "Here's a graph of the two lines\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28600%2C600%2C-15%2C15%2C-15%2C15%2C+-3x-12%2C+2x%2B12%29 ALIGN=MIDDLE  ALT=\"graph%28600%2C600%2C-15%2C15%2C-15%2C15%2C+-3x-12%2C+2x%2B12%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The intersection between these two lines is given below using substituion:\r<BR>\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=linear_substitution.solver>SOLVE linear system by SUBSTITUTION</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT>Solve:<BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+system%28+%0D%0A++++3%5Cx+%2B+1%5Cy+=+-12%2C%0D%0A++++-2%5Cx+%2B+1%5Cy+=+12+%29%0D%0A++ ALIGN=MIDDLE  ALT=\"+system%28+%0D%0A++++3%5Cx+%2B+1%5Cy+=+-12%2C%0D%0A++++-2%5Cx+%2B+1%5Cy+=+12+%29%0D%0A++\"   DISABLED_style= \"max-width:90%; height:auto;\" >We'll use substitution. After moving 1*y to the right, we get:<BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2Ax+=+-12+-+1%2Ay ALIGN=MIDDLE  ALT=\"3%2Ax+=+-12+-+1%2Ay\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-12%2F3+-+1%2Ay%2F3 ALIGN=MIDDLE  ALT=\"x+=+-12%2F3+-+1%2Ay%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Substitute that<BR>\n" );
document.write( "  into another equation:<BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2%2A%28-12%2F3+-+1%2Ay%2F3%29+%2B+1%5Cy+=+12 ALIGN=MIDDLE  ALT=\"-2%2A%28-12%2F3+-+1%2Ay%2F3%29+%2B+1%5Cy+=+12\"   DISABLED_style= \"max-width:90%; height:auto;\" > and simplify: So, we know that y=2.4. Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-12%2F3+-+1%2Ay%2F3 ALIGN=MIDDLE  ALT=\"x+=+-12%2F3+-+1%2Ay%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, x=-4.8.<BR>\n" );
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document.write( "  Answer: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28+x=-4.8%2C+y=2.4+%29 ALIGN=MIDDLE  ALT=\"system%28+x=-4.8%2C+y=2.4+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
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document.write( "Same results as before.\r<BR>\n" );
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document.write( "I hope the above steps and explanation were helpful. \r<BR>\n" );
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document.write( "For Step-By-Step videos on Introduction to Algebra, please visit http://www.FreedomUniversity.TV/courses/IntroAlgebra and for Trigonometry please visit http://www.FreedomUniversity.TV/courses/Trigonometry. \r<BR>\n" );
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document.write( "Also, good luck in your studies and contact me at john@e-liteworks.com for your future math needs.\r<BR>\n" );
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document.write( "Respectfully, <BR>\n" );
document.write( "Dr J\r<BR>\n" );
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document.write( "http://www.FreedomUniversity.TV </SPAN>\n" );
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