document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=389130>Question 389130</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 1.Solve each system of equations using the elimination method.11x + 16y = 922x + 3y = 18\r<BR>\n" );
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document.write( "2.Solve each system of equations using the substitution method. 4x - y = 32y = -2 x + 70\r<BR>\n" );
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document.write( "3.Rewrite the linear programming problem as a maximization problem with constraints involving inequalities of the form&#8804; a constant (with the exception of the inequalities x&#8805;0, y&#8805;0, and z &#8805;0).<BR>\n" );
document.write( "Minimize   C = 2x-3y<BR>\n" );
document.write( "Subject to      3x +5y &#8805;20<BR>\n" );
document.write( "                          3x+y&#8804;16<BR>\n" );
document.write( "                          -2x+y&#8804;1<BR>\n" );
document.write( "                          x&#8805;0, y&#8805;0\r<BR>\n" );
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document.write( "4.  Find the distance between P and Q.P ( -8 , -9 ) and Q ( 10 , -9 ) \r<BR>\n" );
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document.write( "5.Find the distance between P and Q. P ( -9 , -5 ) Q ( 9 , -21 ) \r<BR>\n" );
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document.write( "6.Write an equation of the line that passes through the given two points ( -2 , 7 ), ( 4 , -23 ) <BR>\n" );
document.write( "	 \r<BR>\n" );
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document.write( "*[invoke calculating_slope -2, 7, 4, -23]\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #275650 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20850)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=275650\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\">     \r<BR>\n" );
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document.write( "1.Solve each system of equations using the elimination method.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=11x+%2B+16y+=+92 ALIGN=MIDDLE  ALT=\"11x+%2B+16y+=+92\"   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+3y+=+18 ALIGN=MIDDLE  ALT=\"2x+%2B+3y+=+18\"   DISABLED_style= \"max-width:90%; height:auto;\" > .....I guess this is your system....\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=solving_linear_system_by_elimination.solver>Solving a System of Linear Equations by Elimination/Addition</A></TD></TR>\n" );
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document.write( "  Lets start with the given system of linear equations<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=11%2Ax%2B16%2Ay=92 ALIGN=MIDDLE  ALT=\"11%2Ax%2B16%2Ay=92\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Ax%2B3%2Ay=18 ALIGN=MIDDLE  ALT=\"2%2Ax%2B3%2Ay=18\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  In order to solve for one variable, we must eliminate the other variable. So if we wanted to solve for y, we would have to eliminate x (or vice versa).<BR>\n" );
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document.write( "  So lets eliminate x. In order to do that, we need to have both x coefficients that are equal but have opposite signs (for instance 2 and -2 are equal but have opposite signs). This way they will add to zero.<BR>\n" );
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document.write( "  So to make the x coefficients equal but opposite, we need to multiply both x coefficients by some number to get them to an equal number. So if we wanted to get 11 and 2 to some equal number, we could try to get them to the LCM.<BR>\n" );
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document.write( "  Since the LCM of 11 and 2 is 22, we need to multiply both sides of the top equation by 2 and multiply both sides of the bottom equation by -11 like this:<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2A%2811%2Ax%2B16%2Ay%29=%2892%29%2A2 ALIGN=MIDDLE  ALT=\"2%2A%2811%2Ax%2B16%2Ay%29=%2892%29%2A2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply the top equation (both sides) by 2<BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-11%2A%282%2Ax%2B3%2Ay%29=%2818%29%2A-11 ALIGN=MIDDLE  ALT=\"-11%2A%282%2Ax%2B3%2Ay%29=%2818%29%2A-11\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply the bottom equation (both sides) by -11<BR>\n" );
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document.write( "  So after multiplying we get this:<BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=22%2Ax%2B32%2Ay=184 ALIGN=MIDDLE  ALT=\"22%2Ax%2B32%2Ay=184\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-22%2Ax-33%2Ay=-198 ALIGN=MIDDLE  ALT=\"-22%2Ax-33%2Ay=-198\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  Notice how 22 and -22 add to zero (ie <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=22%2B-22=0 ALIGN=MIDDLE  ALT=\"22%2B-22=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
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document.write( "  <BR>\n" );
document.write( "  Now add the equations together. In order to add 2 equations, group like terms and combine them<BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2822%2Ax-22%2Ax%29%2B%2832%2Ay-33%2Ay%29=184-198 ALIGN=MIDDLE  ALT=\"%2822%2Ax-22%2Ax%29%2B%2832%2Ay-33%2Ay%29=184-198\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2822-22%29%2Ax%2B%2832-33%29y=184-198 ALIGN=MIDDLE  ALT=\"%2822-22%29%2Ax%2B%2832-33%29y=184-198\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cross%2822%2B-22%29%2Ax%2B%2832-33%29%2Ay=184-198 ALIGN=MIDDLE  ALT=\"cross%2822%2B-22%29%2Ax%2B%2832-33%29%2Ay=184-198\"   DISABLED_style= \"max-width:90%; height:auto;\" > Notice the x coefficients add to zero and cancel out. This means we've eliminated x altogether.<BR>\n" );
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document.write( "  So after adding and canceling out the x terms we're left with:<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2Ay=-14 ALIGN=MIDDLE  ALT=\"-1%2Ay=-14\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-14%2F-1 ALIGN=MIDDLE  ALT=\"y=-14%2F-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1 ALIGN=MIDDLE  ALT=\"-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > to solve for y<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=14 ALIGN=MIDDLE  ALT=\"y=14\"   DISABLED_style= \"max-width:90%; height:auto;\" > Reduce<BR>\n" );
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document.write( "  <BR>\n" );
document.write( "  Now plug this answer into the top equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=11%2Ax%2B16%2Ay=92 ALIGN=MIDDLE  ALT=\"11%2Ax%2B16%2Ay=92\"   DISABLED_style= \"max-width:90%; height:auto;\" > to solve for x<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=11%2Ax%2B16%2814%29=92 ALIGN=MIDDLE  ALT=\"11%2Ax%2B16%2814%29=92\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=14 ALIGN=MIDDLE  ALT=\"y=14\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=11%2Ax%2B224=92 ALIGN=MIDDLE  ALT=\"11%2Ax%2B224=92\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=11%2Ax=92-224 ALIGN=MIDDLE  ALT=\"11%2Ax=92-224\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=224 ALIGN=MIDDLE  ALT=\"224\"   DISABLED_style= \"max-width:90%; height:auto;\" > from both sides<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=11%2Ax=-132 ALIGN=MIDDLE  ALT=\"11%2Ax=-132\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine the terms on the right side<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cross%28%281%2F11%29%2811%29%29%2Ax=%28-132%29%281%2F11%29 ALIGN=MIDDLE  ALT=\"cross%28%281%2F11%29%2811%29%29%2Ax=%28-132%29%281%2F11%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply both sides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F11 ALIGN=MIDDLE  ALT=\"1%2F11\"   DISABLED_style= \"max-width:90%; height:auto;\" >. This will cancel out <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=11 ALIGN=MIDDLE  ALT=\"11\"   DISABLED_style= \"max-width:90%; height:auto;\" > on the left side.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-12 ALIGN=MIDDLE  ALT=\"x=-12\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply the terms on the right side<BR>\n" );
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document.write( "  <BR>\n" );
document.write( "  So our answer is <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-12 ALIGN=MIDDLE  ALT=\"x=-12\"   DISABLED_style= \"max-width:90%; height:auto;\" >,  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=14 ALIGN=MIDDLE  ALT=\"y=14\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  which also looks like<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-12 ALIGN=MIDDLE  ALT=\"-12\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=14 ALIGN=MIDDLE  ALT=\"14\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Notice if we graph the equations (if you need help with graphing, check out this <a  rel=nofollow HREF=http://www.algebra.com/algebra/homework/Linear-equations/graphing-linear-equations.solver>solver</a>)<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=11%2Ax%2B16%2Ay=92 ALIGN=MIDDLE  ALT=\"11%2Ax%2B16%2Ay=92\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Ax%2B3%2Ay=18 ALIGN=MIDDLE  ALT=\"2%2Ax%2B3%2Ay=18\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  we get <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%0D%0A++drawing%28+500%2C+600%2C+-22%2C+22%2C+-24%2C+24%2C%0D%0A++++graph%28+500%2C+600%2C+-22%2C+22%2C+-24%2C+24%2C+%2892-11%2Ax%29%2F16%2C+%2818-2%2Ax%29%2F3+%29%2C%0D%0A++++blue%28++circle%28+-12%2C+14%2C+0.293333333333333+%29+%29+%0D%0A++%29%0D%0A++ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=11%2Ax%2B16%2Ay=92 ALIGN=MIDDLE  ALT=\"11%2Ax%2B16%2Ay=92\"   DISABLED_style= \"max-width:90%; height:auto;\" > (red) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Ax%2B3%2Ay=18 ALIGN=MIDDLE  ALT=\"2%2Ax%2B3%2Ay=18\"   DISABLED_style= \"max-width:90%; height:auto;\" > (green) (hint: you may have to solve for y to graph these) and the intersection of the lines (blue circle).<BR>\n" );
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document.write( "  <BR>\n" );
document.write( "  and we can see that the two equations intersect at (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-12 ALIGN=MIDDLE  ALT=\"-12\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=14 ALIGN=MIDDLE  ALT=\"14\"   DISABLED_style= \"max-width:90%; height:auto;\" >). This verifies our answer.</TD></TR>\n" );
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document.write( "2.Solve each system of equations using the substitution method. \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x+-+y+=+32 ALIGN=MIDDLE  ALT=\"4x+-+y+=+32\"   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+=+-2+x+%2B+70+ ALIGN=MIDDLE  ALT=\"y+=+-2+x+%2B+70+\"   DISABLED_style= \"max-width:90%; height:auto;\" >..or..<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2+x+%2B+y+=70+ ALIGN=MIDDLE  ALT=\"2+x+%2B+y+=70+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\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++++4%5Cx+%2B+-1%5Cy+=+32%2C%0D%0A++++2%5Cx+%2B+1%5Cy+=+70+%29%0D%0A++ ALIGN=MIDDLE  ALT=\"+system%28+%0D%0A++++4%5Cx+%2B+-1%5Cy+=+32%2C%0D%0A++++2%5Cx+%2B+1%5Cy+=+70+%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=4%2Ax+=+32+-+-1%2Ay ALIGN=MIDDLE  ALT=\"4%2Ax+=+32+-+-1%2Ay\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+32%2F4+-+-1%2Ay%2F4 ALIGN=MIDDLE  ALT=\"x+=+32%2F4+-+-1%2Ay%2F4\"   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%2832%2F4+-+-1%2Ay%2F4%29+%2B+1%5Cy+=+70 ALIGN=MIDDLE  ALT=\"2%2A%2832%2F4+-+-1%2Ay%2F4%29+%2B+1%5Cy+=+70\"   DISABLED_style= \"max-width:90%; height:auto;\" > and simplify: So, we know that y=36. Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+32%2F4+-+-1%2Ay%2F4 ALIGN=MIDDLE  ALT=\"x+=+32%2F4+-+-1%2Ay%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" >, x=17.<BR>\n" );
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document.write( "  Answer: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28+x=17%2C+y=36+%29 ALIGN=MIDDLE  ALT=\"system%28+x=17%2C+y=36+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
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document.write( "4. Find the distance between P and Q.P ( -8 , -9 ) and Q ( 10 , -9 ) \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=Distance_between_a_point_and_a_line.solver>Finding a distance between a point given by coordinates (x, y)  and a line given by equation y=ax+b</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT>We want to find the perpendicular distance between a point given by coordinates (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-8 ALIGN=MIDDLE  ALT=\"-8\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-9 ALIGN=MIDDLE  ALT=\"-9\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
document.write( "  and a line given by equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=10%2Ax%2B-9 ALIGN=MIDDLE  ALT=\"y=10%2Ax%2B-9\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  First, let's draw a diagram of general situation with point <b>P</b> (xo, yo) and <BR>\n" );
document.write( "  line <b>L:</b> y= a.x + b. The required distance is <b>PC</b>. (in the diagram below)<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28+200%2C+200%2C+-2%2C+8%2C+-2%2C+8%2C+line%28-2%2C-2%2C8%2C8%29%2Cline%280%2C0%2C0%2C4%29%2Cline%280%2C4%2C4%2C4%29%2Cgreen%28line%280%2C4%2C2%2C2%29%29%2Clocate%280%2C5%2CP%28xo%2Cyo%29%29%2Clocate%280%2C0%2CA%28x1%2Cy1%29%29%2C+locate%284%2C4%2CB%28x2%2Cy2%29%29%2Clocate%282%2C2%2CC%29%2Clocate%286%2C6%2CL%29%2C++red%28circle%280%2C0%2C.1%29%29%2Cred%28circle%284%2C4%2C.1%29%29+%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <b>Methodology</b><BR>\n" );
document.write( "  We will first find the vertices of the triangle in order to get the side lengths and then by applying <BR>\n" );
document.write( "  <b><A  rel=nofollow HREF=Sine_rule.wikipedia>Sine Rule</A></b> on <A  rel=nofollow HREF=Special_right_triangles.wikipedia>right angle triangle</A> <b>PAB</b> and <b>PBC</b> we will calculate the desired distance <b>PC</b>. <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <b>Step1</b><BR>\n" );
document.write( "  Calculation of the vertices of triangle PAB:<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Draw a vertical line passing through the point 'P'. This line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-8 ALIGN=MIDDLE  ALT=\"x=-8\"   DISABLED_style= \"max-width:90%; height:auto;\" > will cut the given line 'L' <BR>\n" );
document.write( "  at point 'A'. The X coordinate of A(<b>x1</b>) will be same as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=xo=-8 ALIGN=MIDDLE  ALT=\"xo=-8\"   DISABLED_style= \"max-width:90%; height:auto;\" >. To find the Y-coordinate of<BR>\n" );
document.write( "  'A' we will use the fact that <b>point 'A' lies on the given line 'L' and satisfies the equation <BR>\n" );
document.write( "  of the line 'L' </b>.<BR>\n" );
document.write( "  Now, plug this <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x1=-8 ALIGN=MIDDLE  ALT=\"x1=-8\"   DISABLED_style= \"max-width:90%; height:auto;\" > in to the equation of line: y=10*x+-9 <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y1=10%2A-8+%2B-9 ALIGN=MIDDLE  ALT=\"y1=10%2A-8+%2B-9\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y1=-89 ALIGN=MIDDLE  ALT=\"y1=-89\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Hence, Point (A)(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x1=-8 ALIGN=MIDDLE  ALT=\"x1=-8\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y1=-89 ALIGN=MIDDLE  ALT=\"y1=-89\"   DISABLED_style= \"max-width:90%; height:auto;\" >) <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Similarly, <BR>\n" );
document.write( "  Draw a horizontal line passing through the point 'P'. This line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-9 ALIGN=MIDDLE  ALT=\"y=-9\"   DISABLED_style= \"max-width:90%; height:auto;\" > will cut the given line 'L' <BR>\n" );
document.write( "  at point 'B'. The Y coordinate of B(<b>y2</b>) will be same as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=yo=-9 ALIGN=MIDDLE  ALT=\"yo=-9\"   DISABLED_style= \"max-width:90%; height:auto;\" >. To find the X-coordinate of<BR>\n" );
document.write( "  B we will use the fact that <b>point 'B' lies on the given line 'L' and satisfies the equation <BR>\n" );
document.write( "  of the line 'L' </b>.<BR>\n" );
document.write( "  Now, plug this <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y2=-9 ALIGN=MIDDLE  ALT=\"y2=-9\"   DISABLED_style= \"max-width:90%; height:auto;\" > in to the equation of line: y=10*x+-9<BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-9=10%2Ax2%2B-9 ALIGN=MIDDLE  ALT=\"-9=10%2Ax2%2B-9\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x2=+%28-9--9%29%2F10 ALIGN=MIDDLE  ALT=\"x2=+%28-9--9%29%2F10\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x2=0 ALIGN=MIDDLE  ALT=\"x2=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Hence, Point (B)(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x2=0 ALIGN=MIDDLE  ALT=\"x2=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y2=-9 ALIGN=MIDDLE  ALT=\"y2=-9\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Now, we have all the vertices of the triangle PAB<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <b>Step2</b><BR>\n" );
document.write( "  Calculation of the side lengths using <A  rel=nofollow HREF=Distance_formula.wikipedia>distance formula</A>: <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=sqrt%28%28x1-x2%29%5E2+%2B+%28y1-y2%29%5E2%29 ALIGN=MIDDLE  ALT=\"d=sqrt%28%28x1-x2%29%5E2+%2B+%28y1-y2%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Hence, The side lengths PA, PB and AB are<BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=PA=80 ALIGN=MIDDLE  ALT=\"PA=80\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=PB=8 ALIGN=MIDDLE  ALT=\"PB=8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AB=80.3990049689671 ALIGN=MIDDLE  ALT=\"AB=80.3990049689671\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "   <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <b>Step3</b><BR>\n" );
document.write( "  Apply <b>Sine rule</b> on common angle <b>B</b> in triangle <b>PAB</b> and triangle <b>PBC</b>.<BR>\n" );
document.write( "  Both triangle <b>PAB</b> and triangle <b>PBC</b> are <A  rel=nofollow HREF=Special_right_triangles.wikipedia>right angle triangle</A> and points 'A', 'B' and 'C' lay on the given line L. <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Sin%28B%29=+AP%2FAB=PC%2FBP ALIGN=MIDDLE  ALT=\"Sin%28B%29=+AP%2FAB=PC%2FBP\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=PC=%28AP%2ABP%29%2FAB=+7.96029752167991 ALIGN=MIDDLE  ALT=\"PC=%28AP%2ABP%29%2FAB=+7.96029752167991\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  PC is the required perpendicular distance of the point <b>P</b> (-8, -9) from line given <BR>\n" );
document.write( "  line<b>L1:</b> y=10*x+-9.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  For better understanding of this concept, look at the Lesson based on the above concept. <BR>\n" );
document.write( "   <A  rel=nofollow HREF=http://www.algebra.com/algebra/homework/Coordinate-system/Distance-of-a-Point-From-a-Line.lesson> Lesson </A></TD></TR>\n" );
document.write( "</TABLE><BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "5.Find the distance between P and Q. P ( -9 , -5 ) Q ( 9 , -21 ) \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<TABLE  BORDER=1>\n" );
document.write( "\n" );
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=Distance_between_a_point_and_a_line.solver>Finding a distance between a point given by coordinates (x, y)  and a line given by equation y=ax+b</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT>We want to find the perpendicular distance between a point given by coordinates (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-9 ALIGN=MIDDLE  ALT=\"-9\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-5 ALIGN=MIDDLE  ALT=\"-5\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
document.write( "  and a line given by equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=9%2Ax%2B-21 ALIGN=MIDDLE  ALT=\"y=9%2Ax%2B-21\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  First, let's draw a diagram of general situation with point <b>P</b> (xo, yo) and <BR>\n" );
document.write( "  line <b>L:</b> y= a.x + b. The required distance is <b>PC</b>. (in the diagram below)<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28+200%2C+200%2C+-2%2C+8%2C+-2%2C+8%2C+line%28-2%2C-2%2C8%2C8%29%2Cline%280%2C0%2C0%2C4%29%2Cline%280%2C4%2C4%2C4%29%2Cgreen%28line%280%2C4%2C2%2C2%29%29%2Clocate%280%2C5%2CP%28xo%2Cyo%29%29%2Clocate%280%2C0%2CA%28x1%2Cy1%29%29%2C+locate%284%2C4%2CB%28x2%2Cy2%29%29%2Clocate%282%2C2%2CC%29%2Clocate%286%2C6%2CL%29%2C++red%28circle%280%2C0%2C.1%29%29%2Cred%28circle%284%2C4%2C.1%29%29+%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <b>Methodology</b><BR>\n" );
document.write( "  We will first find the vertices of the triangle in order to get the side lengths and then by applying <BR>\n" );
document.write( "  <b><A  rel=nofollow HREF=Sine_rule.wikipedia>Sine Rule</A></b> on <A  rel=nofollow HREF=Special_right_triangles.wikipedia>right angle triangle</A> <b>PAB</b> and <b>PBC</b> we will calculate the desired distance <b>PC</b>. <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <b>Step1</b><BR>\n" );
document.write( "  Calculation of the vertices of triangle PAB:<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Draw a vertical line passing through the point 'P'. This line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-9 ALIGN=MIDDLE  ALT=\"x=-9\"   DISABLED_style= \"max-width:90%; height:auto;\" > will cut the given line 'L' <BR>\n" );
document.write( "  at point 'A'. The X coordinate of A(<b>x1</b>) will be same as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=xo=-9 ALIGN=MIDDLE  ALT=\"xo=-9\"   DISABLED_style= \"max-width:90%; height:auto;\" >. To find the Y-coordinate of<BR>\n" );
document.write( "  'A' we will use the fact that <b>point 'A' lies on the given line 'L' and satisfies the equation <BR>\n" );
document.write( "  of the line 'L' </b>.<BR>\n" );
document.write( "  Now, plug this <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x1=-9 ALIGN=MIDDLE  ALT=\"x1=-9\"   DISABLED_style= \"max-width:90%; height:auto;\" > in to the equation of line: y=9*x+-21 <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y1=9%2A-9+%2B-21 ALIGN=MIDDLE  ALT=\"y1=9%2A-9+%2B-21\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y1=-102 ALIGN=MIDDLE  ALT=\"y1=-102\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Hence, Point (A)(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x1=-9 ALIGN=MIDDLE  ALT=\"x1=-9\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y1=-102 ALIGN=MIDDLE  ALT=\"y1=-102\"   DISABLED_style= \"max-width:90%; height:auto;\" >) <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Similarly, <BR>\n" );
document.write( "  Draw a horizontal line passing through the point 'P'. This line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-5 ALIGN=MIDDLE  ALT=\"y=-5\"   DISABLED_style= \"max-width:90%; height:auto;\" > will cut the given line 'L' <BR>\n" );
document.write( "  at point 'B'. The Y coordinate of B(<b>y2</b>) will be same as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=yo=-5 ALIGN=MIDDLE  ALT=\"yo=-5\"   DISABLED_style= \"max-width:90%; height:auto;\" >. To find the X-coordinate of<BR>\n" );
document.write( "  B we will use the fact that <b>point 'B' lies on the given line 'L' and satisfies the equation <BR>\n" );
document.write( "  of the line 'L' </b>.<BR>\n" );
document.write( "  Now, plug this <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y2=-5 ALIGN=MIDDLE  ALT=\"y2=-5\"   DISABLED_style= \"max-width:90%; height:auto;\" > in to the equation of line: y=9*x+-21<BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-5=9%2Ax2%2B-21 ALIGN=MIDDLE  ALT=\"-5=9%2Ax2%2B-21\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x2=+%28-5--21%29%2F9 ALIGN=MIDDLE  ALT=\"x2=+%28-5--21%29%2F9\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x2=1.77777777777778 ALIGN=MIDDLE  ALT=\"x2=1.77777777777778\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Hence, Point (B)(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x2=1.77777777777778 ALIGN=MIDDLE  ALT=\"x2=1.77777777777778\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y2=-5 ALIGN=MIDDLE  ALT=\"y2=-5\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Now, we have all the vertices of the triangle PAB<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <b>Step2</b><BR>\n" );
document.write( "  Calculation of the side lengths using <A  rel=nofollow HREF=Distance_formula.wikipedia>distance formula</A>: <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=sqrt%28%28x1-x2%29%5E2+%2B+%28y1-y2%29%5E2%29 ALIGN=MIDDLE  ALT=\"d=sqrt%28%28x1-x2%29%5E2+%2B+%28y1-y2%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Hence, The side lengths PA, PB and AB are<BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=PA=97 ALIGN=MIDDLE  ALT=\"PA=97\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=PB=10.7777777777778 ALIGN=MIDDLE  ALT=\"PB=10.7777777777778\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=AB=97.5969287110366 ALIGN=MIDDLE  ALT=\"AB=97.5969287110366\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "   <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <b>Step3</b><BR>\n" );
document.write( "  Apply <b>Sine rule</b> on common angle <b>B</b> in triangle <b>PAB</b> and triangle <b>PBC</b>.<BR>\n" );
document.write( "  Both triangle <b>PAB</b> and triangle <b>PBC</b> are <A  rel=nofollow HREF=Special_right_triangles.wikipedia>right angle triangle</A> and points 'A', 'B' and 'C' lay on the given line L. <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Sin%28B%29=+AP%2FAB=PC%2FBP ALIGN=MIDDLE  ALT=\"Sin%28B%29=+AP%2FAB=PC%2FBP\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=PC=%28AP%2ABP%29%2FAB=+10.7118580292601 ALIGN=MIDDLE  ALT=\"PC=%28AP%2ABP%29%2FAB=+10.7118580292601\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  PC is the required perpendicular distance of the point <b>P</b> (-9, -5) from line given <BR>\n" );
document.write( "  line<b>L1:</b> y=9*x+-21.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  For better understanding of this concept, look at the Lesson based on the above concept. <BR>\n" );
document.write( "   <A  rel=nofollow HREF=http://www.algebra.com/algebra/homework/Coordinate-system/Distance-of-a-Point-From-a-Line.lesson> Lesson </A></TD></TR>\n" );
document.write( "</TABLE><BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "6.Write an equation of the line that passes through the given two points <BR>\n" );
document.write( "( -2 , 7 ), ( 4 , -23 ) \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<TABLE  BORDER=1>\n" );
document.write( "\n" );
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=calculating_slope.solver>Finding the Equation of a Line</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT>First lets find the slope through the points (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2 ALIGN=MIDDLE  ALT=\"-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7 ALIGN=MIDDLE  ALT=\"7\"   DISABLED_style= \"max-width:90%; height:auto;\" >) and (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-23 ALIGN=MIDDLE  ALT=\"-23\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
document.write( "  <BR>\n" );
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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;\" > Start with the slope formula (note: (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B1%5D ALIGN=MIDDLE  ALT=\"x%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B1%5D ALIGN=MIDDLE  ALT=\"y%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >) is the first point (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2 ALIGN=MIDDLE  ALT=\"-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7 ALIGN=MIDDLE  ALT=\"7\"   DISABLED_style= \"max-width:90%; height:auto;\" >) and  (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B2%5D ALIGN=MIDDLE  ALT=\"x%5B2%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B2%5D ALIGN=MIDDLE  ALT=\"y%5B2%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >) is the second point (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-23 ALIGN=MIDDLE  ALT=\"-23\"   DISABLED_style= \"max-width:90%; height:auto;\" >))<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=%28-23-7%29%2F%284--2%29 ALIGN=MIDDLE  ALT=\"m=%28-23-7%29%2F%284--2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B2%5D=-23 ALIGN=MIDDLE  ALT=\"y%5B2%5D=-23\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B1%5D=7 ALIGN=MIDDLE  ALT=\"y%5B1%5D=7\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B2%5D=4 ALIGN=MIDDLE  ALT=\"x%5B2%5D=4\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B1%5D=-2 ALIGN=MIDDLE  ALT=\"x%5B1%5D=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >  (these are the coordinates of given points)<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=+-30%2F6 ALIGN=MIDDLE  ALT=\"m=+-30%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract the terms in the numerator <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-23-7 ALIGN=MIDDLE  ALT=\"-23-7\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-30 ALIGN=MIDDLE  ALT=\"-30\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Subtract the terms in the denominator <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4--2 ALIGN=MIDDLE  ALT=\"4--2\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6 ALIGN=MIDDLE  ALT=\"6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=-5 ALIGN=MIDDLE  ALT=\"m=-5\"   DISABLED_style= \"max-width:90%; height:auto;\" > Reduce<BR>\n" );
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document.write( "  So the slope is<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=-5 ALIGN=MIDDLE  ALT=\"m=-5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  ------------------------------------------------<BR>\n" );
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document.write( "Now let's use the point-slope formula to find the equation of the line:<BR>\n" );
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document.write( "  ------Point-Slope Formula------<BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-y%5B1%5D=m%28x-x%5B1%5D%29 ALIGN=MIDDLE  ALT=\"y-y%5B1%5D=m%28x-x%5B1%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m ALIGN=MIDDLE  ALT=\"m\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the slope, and (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B1%5D ALIGN=MIDDLE  ALT=\"x%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B1%5D ALIGN=MIDDLE  ALT=\"y%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >) is one of the given points<BR>\n" );
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document.write( "  So lets use the Point-Slope Formula to find the equation of the line<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-7=%28-5%29%28x--2%29 ALIGN=MIDDLE  ALT=\"y-7=%28-5%29%28x--2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=-5 ALIGN=MIDDLE  ALT=\"m=-5\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B1%5D=-2 ALIGN=MIDDLE  ALT=\"x%5B1%5D=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B1%5D=7 ALIGN=MIDDLE  ALT=\"y%5B1%5D=7\"   DISABLED_style= \"max-width:90%; height:auto;\" > (these values are given)<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-7=%28-5%29%28x%2B2%29 ALIGN=MIDDLE  ALT=\"y-7=%28-5%29%28x%2B2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Rewrite <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;\" > as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B2 ALIGN=MIDDLE  ALT=\"x%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-7=-5x%2B%28-5%29%282%29 ALIGN=MIDDLE  ALT=\"y-7=-5x%2B%28-5%29%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Distribute <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-5 ALIGN=MIDDLE  ALT=\"-5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-7=-5x-10 ALIGN=MIDDLE  ALT=\"y-7=-5x-10\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-5 ALIGN=MIDDLE  ALT=\"-5\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-10%2F1 ALIGN=MIDDLE  ALT=\"-10%2F1\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Now reduce <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-10%2F1 ALIGN=MIDDLE  ALT=\"-10%2F1\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-10 ALIGN=MIDDLE  ALT=\"-10\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-5x-10%2B7 ALIGN=MIDDLE  ALT=\"y=-5x-10%2B7\"   DISABLED_style= \"max-width:90%; height:auto;\" > Add <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7 ALIGN=MIDDLE  ALT=\"7\"   DISABLED_style= \"max-width:90%; height:auto;\" > to  both sides to isolate y<BR>\n" );
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document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-5x-3 ALIGN=MIDDLE  ALT=\"y=-5x-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-10 ALIGN=MIDDLE  ALT=\"-10\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7 ALIGN=MIDDLE  ALT=\"7\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3 ALIGN=MIDDLE  ALT=\"-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
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document.write( "  ------------------------------------------------------------------------------------------------------------<BR>\n" );
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document.write( "  Answer:<BR>\n" );
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document.write( "  So the equation of the line which goes through the points (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2 ALIGN=MIDDLE  ALT=\"-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7 ALIGN=MIDDLE  ALT=\"7\"   DISABLED_style= \"max-width:90%; height:auto;\" >) and (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-23 ALIGN=MIDDLE  ALT=\"-23\"   DISABLED_style= \"max-width:90%; height:auto;\" >)  is:<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-5x-3 ALIGN=MIDDLE  ALT=\"y=-5x-3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  The equation is now in <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;\" > form (which is slope-intercept form) where the slope is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=-5 ALIGN=MIDDLE  ALT=\"m=-5\"   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 ALIGN=MIDDLE  ALT=\"b=-3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  <BR>\n" );
document.write( "  Notice if we graph the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-5x-3 ALIGN=MIDDLE  ALT=\"y=-5x-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > and plot the points (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2 ALIGN=MIDDLE  ALT=\"-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7 ALIGN=MIDDLE  ALT=\"7\"   DISABLED_style= \"max-width:90%; height:auto;\" >) and (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-23 ALIGN=MIDDLE  ALT=\"-23\"   DISABLED_style= \"max-width:90%; height:auto;\" >),  we get this: (note: if you need help with graphing, check out this <a  rel=nofollow HREF=http://www.algebra.com/algebra/homework/Linear-equations/graphing-linear-equations.solver>solver<a>)<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28500%2C+500%2C+-8%2C+10%2C+-17%2C+1%2C%0D%0A++graph%28500%2C+500%2C+-8%2C+10%2C+-17%2C+1%2C%28-5%29x%2B-3%29%2C%0D%0A++circle%28-2%2C7%2C0.12%29%2C%0D%0A++circle%28-2%2C7%2C0.12%2B0.03%29%2C%0D%0A++circle%284%2C-23%2C0.12%29%2C%0D%0A++circle%284%2C-23%2C0.12%2B0.03%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=-5x-3 ALIGN=MIDDLE  ALT=\"y=-5x-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > through the points (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2 ALIGN=MIDDLE  ALT=\"-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7 ALIGN=MIDDLE  ALT=\"7\"   DISABLED_style= \"max-width:90%; height:auto;\" >) and (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-23 ALIGN=MIDDLE  ALT=\"-23\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
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document.write( "  Notice how the two points lie on the line. This graphically verifies our answer.<BR>\n" );
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document.write( "  </TD></TR>\n" );
document.write( "</TABLE><BR>\n" );
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document.write( " </SPAN>\n" );
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