document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=117570>Question 117570</A>: <I><SPAN STYLE=\"word-break: break-all;\"> linear combinations????? \r<BR>\n" );
document.write( "\n" );
document.write( "i don't understand any of this i'm not looking for the answer as i know it but i'm looking for step by step help with the work. plus they have added THREE variables now and i didn't even understand two so please help me.\r<BR>\n" );
document.write( "\n" );
document.write( "4x + y = -12<BR>\n" );
document.write( "2x + 2y = -15? (answer -1.5, -6)then with THREE 2a + b =5c = -21<BR>\n" );
document.write( "                                                 a + 2b - 2c = -15<BR>\n" );
document.write( "                                                 a - 4b + c = 18 (answer -1, -4, 3 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #85565 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=85565\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> 1) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x+%2B+y+=+-12 ALIGN=MIDDLE  ALT=\"4x+%2B+y+=+-12\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "2) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x+%2B+2y=+-15 ALIGN=MIDDLE  ALT=\"2x+%2B+2y=+-15\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "There are a number of ways to solve these, so let's look at all of them, step by step.  First, remember that an equation in two first degree variables is a description of a straight line on the xy plane.  If you have two lines, the point where they intersect (if they do intersect) is an x, y ordered pair that satisfies both equations at the same time.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "First, the <b>Substitution Method:</b>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Step 1:  Solve one of the equations for one of the variables.  We'll solve equation 1) for y.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x+%2B+y+=+-12 ALIGN=MIDDLE  ALT=\"4x+%2B+y+=+-12\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-4x-12 ALIGN=MIDDLE  ALT=\"y=-4x-12\"   DISABLED_style= \"max-width:90%; height:auto;\" >  Now we have an expression for y that can be substituted for y in equation 2), giving us a single equation in a single variable.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Step 2: Substitute the expression for y in equation 2)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x+%2B+2%28-4x-12%29=+-15 ALIGN=MIDDLE  ALT=\"2x+%2B+2%28-4x-12%29=+-15\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Step 3: Solve for x<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x-8x-24=-15 ALIGN=MIDDLE  ALT=\"2x-8x-24=-15\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-6x=9 ALIGN=MIDDLE  ALT=\"-6x=9\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-%289%2F6%29=-%283%2F2%29 ALIGN=MIDDLE  ALT=\"x=-%289%2F6%29=-%283%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Step 4: Substitute this new found value for x into the form of equation 1) where we solved for y:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-4%28-3%2F2%29-12 ALIGN=MIDDLE  ALT=\"y=-4%28-3%2F2%29-12\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<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;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now we know that the two lines represented by the two equations intersect in the point (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3%2F2 ALIGN=MIDDLE  ALT=\"-3%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-6 ALIGN=MIDDLE  ALT=\"-6\"   DISABLED_style= \"max-width:90%; height:auto;\" >).\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Next, <b>Gaussian Elimination:</b>\r<BR>\n" );
document.write( "\n" );
document.write( "Step 1: Put the equations in standard form.  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ax%2Bby=c ALIGN=MIDDLE  ALT=\"ax%2Bby=c\"   DISABLED_style= \"max-width:90%; height:auto;\" >  For this problem, they are already in this form.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Discussion: There are several things you can do, and the order in which you do them depends on the particular problem.  You are trying to get one of the variable coefficients (a or b in the standard form) to be zero and the other one to be 1 so that you will have the value of one of the variables.  The things you can do are:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Multiply an equation by a constant value.<BR>\n" );
document.write( "Add/Subtract the two equations term by term and replace one of the equations with the result.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So let's start with this problem:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "1) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x+%2B+y+=+-12 ALIGN=MIDDLE  ALT=\"4x+%2B+y+=+-12\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "2) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x+%2B+2y=+-15 ALIGN=MIDDLE  ALT=\"2x+%2B+2y=+-15\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Multiply equation 2 by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2F2 ALIGN=MIDDLE  ALT=\"-1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "1) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x+%2B+y+=+-12 ALIGN=MIDDLE  ALT=\"4x+%2B+y+=+-12\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "2) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-x+-+y=+15%2F2 ALIGN=MIDDLE  ALT=\"-x+-+y=+15%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Add Eq 1) and 2) and replace eq 2):\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "1) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x+%2B+y+=+-12 ALIGN=MIDDLE  ALT=\"4x+%2B+y+=+-12\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "2) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x+%2B0y=+-9%2F2 ALIGN=MIDDLE  ALT=\"3x+%2B0y=+-9%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Multiply Eq 2) by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F3 ALIGN=MIDDLE  ALT=\"1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "1) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x+%2B+y+=+-12 ALIGN=MIDDLE  ALT=\"4x+%2B+y+=+-12\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "2) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-3%2F2 ALIGN=MIDDLE  ALT=\"x+=+-3%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now we have a value for x, time to find y.  Multiply Eq 2) by -4:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "1) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x+%2B+y+=+-12 ALIGN=MIDDLE  ALT=\"4x+%2B+y+=+-12\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "2) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4x+=+6 ALIGN=MIDDLE  ALT=\"-4x+=+6\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Add the two equations and replace Eq 1)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "1) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0x+%2B+y+=+-6 ALIGN=MIDDLE  ALT=\"0x+%2B+y+=+-6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "2) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4x+=+6 ALIGN=MIDDLE  ALT=\"-4x+=+6\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "And we now have a value for y.  SURPRISE!  We got the same answer as the Substitution Method  (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-%283%2F2%29 ALIGN=MIDDLE  ALT=\"-%283%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-6 ALIGN=MIDDLE  ALT=\"-6\"   DISABLED_style= \"max-width:90%; height:auto;\" >)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Next, <b>Cramer's Rule</b>\r<BR>\n" );
document.write( "\n" );
document.write( "Step 1:  Create and evaluate the coefficient determinant, D, an array of the coefficients on the variables.  If you have two equations <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%28a%5B1%5Dx%2Bb%5B1%5Dy=c%5B1%5D%29 ALIGN=MIDDLE  ALT=\"red%28a%5B1%5Dx%2Bb%5B1%5Dy=c%5B1%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=green%28a%5B2%5Dx%2Bb%5B2%5Dy=c%5B2%5D%29 ALIGN=MIDDLE  ALT=\"green%28a%5B2%5Dx%2Bb%5B2%5Dy=c%5B2%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then the coefficient determinant is:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D=%28matrix%282%2C2%2Cred%28a%5B1%5D%29%2Cred%28b%5B1%5D%29%2Cgreen%28a%5B2%5D%29%2Cgreen%28b%5B2%5D%29%29%29=red%28a%5B1%5D%29%2Agreen%28b%5B2%5D%29-green%28a%5B2%5D%29%2Ared%28b%5B1%5D%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D=%28matrix%282%2C2%2C4%2C1%2C2%2C2%29%29=4%2A2-2%2A1=6 ALIGN=MIDDLE  ALT=\"D=%28matrix%282%2C2%2C4%2C1%2C2%2C2%29%29=4%2A2-2%2A1=6\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Step 2:  Create the constant matrix.  This is a single column matrix of the 'c' values\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%282%2C1%2Cc%5B1%5D%2Cc%5B2%5D%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%282%2C1%2Cc%5B1%5D%2Cc%5B2%5D%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%282%2C1%2C-12%2C-15%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%282%2C1%2C-12%2C-15%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Step 3:  Create the x Determinant, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D%5Bx%5D ALIGN=MIDDLE  ALT=\"D%5Bx%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > by substituting the constant matrix for the x (first) column in the coefficient matrix, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D ALIGN=MIDDLE  ALT=\"D\"   DISABLED_style= \"max-width:90%; height:auto;\" > and then evaluate.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D%5Bx%5D=%28matrix%282%2C2%2Cred%28c%5B1%5D%29%2Cred%28b%5B1%5D%29%2Cgreen%28c%5B2%5D%29%2Cgreen%28b%5B2%5D%29%29%29=red%28c%5B1%5D%29%2Agreen%28b%5B2%5D%29-green%28c%5B2%5D%29%2Ared%28b%5B1%5D%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D%5Bx%5D=%28matrix%282%2C2%2C-12%2C1%2C-15%2C2%29%29=-12%2A2-%28-15%29%2A1=-9 ALIGN=MIDDLE  ALT=\"D%5Bx%5D=%28matrix%282%2C2%2C-12%2C1%2C-15%2C2%29%29=-12%2A2-%28-15%29%2A1=-9\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Step 4:  Create the y Determinant.  Same as the last step, except that the constant matrix replaces the second column.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D%5By%5D=%28matrix%282%2C2%2Cred%28a%5B1%5D%29%2Cred%28c%5B1%5D%29%2Cgreen%28a%5B2%5D%29%2Cgreen%28c%5B2%5D%29%29%29=red%28a%5B1%5D%29%2Agreen%28c%5B2%5D%29-green%28a%5B2%5D%29%2Ared%28c%5B1%5D%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D%5By%5D=%28matrix%282%2C2%2C4%2C-12%2C2%2C-15%29%29=4%2A%28-15%29-2%2A%28-12%29=-36 ALIGN=MIDDLE  ALT=\"D%5By%5D=%28matrix%282%2C2%2C4%2C-12%2C2%2C-15%29%29=4%2A%28-15%29-2%2A%28-12%29=-36\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Step 5:  Apply Cramer's Rule:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<b>Cramer's Rule</b>:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=D%5Bx%5D%2FD ALIGN=MIDDLE  ALT=\"x=D%5Bx%5D%2FD\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=D%5By%5D%2FD ALIGN=MIDDLE  ALT=\"y=D%5By%5D%2FD\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-9%2F6=-3%2F2 ALIGN=MIDDLE  ALT=\"x=-9%2F6=-3%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-36%2F6=-6 ALIGN=MIDDLE  ALT=\"y=-36%2F6=-6\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "And again we get the same answer. (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-%283%2F2%29 ALIGN=MIDDLE  ALT=\"-%283%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-6 ALIGN=MIDDLE  ALT=\"-6\"   DISABLED_style= \"max-width:90%; height:auto;\" >) I'm feeling pretty confident by now.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The one thing you have to remember with Cramer's rule is that if the coefficient determinant evaluates to zero <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D=0 ALIGN=MIDDLE  ALT=\"D=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > then you either have an inconsistent system (two parallel lines) with no solution set or a consistent and undetermined system of two coincident lines with an infinite solution set.  And you have to use another method to determine which.  The best way is to put your equations into 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;\" > by solving for y and reducing everything to lowest terms.  If the slope values, m, are equal but the intercepts, b, are different, you have two parallel lines.  If they are the same equation, they are the same line quite obviously.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "****************************\r<BR>\n" );
document.write( "\n" );
document.write( "Equations in three variables are just a more complex application of all the same principles that we just discussed.  I'm not going to explain the details, but you should be able to tell what I'm doing.  (You have an equal sign in the wrong place in the first equation, but I'm going to assume you just hit the wrong key and it should be a minus sign)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "First the substitution method:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2a+%2B+b+-5c+=+-21 ALIGN=MIDDLE  ALT=\"2a+%2B+b+-5c+=+-21\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+%2B+2b+-+2c+=+-15 ALIGN=MIDDLE  ALT=\"a+%2B+2b+-+2c+=+-15\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+-+4b+%2B+c+=+18 ALIGN=MIDDLE  ALT=\"a+-+4b+%2B+c+=+18\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=-2b%2B2c-15 ALIGN=MIDDLE  ALT=\"a+=-2b%2B2c-15\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2b%2B2c-15-4b%2Bc=18 ALIGN=MIDDLE  ALT=\"-2b%2B2c-15-4b%2Bc=18\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-6b%2B3c=33 ALIGN=MIDDLE  ALT=\"-6b%2B3c=33\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=2b%2B11 ALIGN=MIDDLE  ALT=\"c=2b%2B11\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=-2b%2B4b%2B22-15 ALIGN=MIDDLE  ALT=\"a=-2b%2B4b%2B22-15\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=2b%2B7 ALIGN=MIDDLE  ALT=\"a=2b%2B7\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%282b%2B7%29%2Bb-5%282b%2B11%29=-21 ALIGN=MIDDLE  ALT=\"2%282b%2B7%29%2Bb-5%282b%2B11%29=-21\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4b%2B14%2Bb-10b-55=-21 ALIGN=MIDDLE  ALT=\"4b%2B14%2Bb-10b-55=-21\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-5b=-21-14%2B55=20 ALIGN=MIDDLE  ALT=\"-5b=-21-14%2B55=20\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<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" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=2%28-4%29%2B7=-1 ALIGN=MIDDLE  ALT=\"a=2%28-4%29%2B7=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=2%28-4%29%2B11=3 ALIGN=MIDDLE  ALT=\"c=2%28-4%29%2B11=3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "Here's Cramer's Rule<BR>\n" );
document.write( "Just remember that you have to repeat the first two columns when you evaluate a 3X3 Determinant\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D=%28matrix%283%2C5%2Cred%28a%5B1%5D%29%2Cred%28b%5B1%5D%29%2Cred%28c%5B1%5D%29%2Cred%28a%5B1%5D%29%2Cred%28b%5B1%5D%29%2Cgreen%28a%5B2%5D%29%2Cgreen%28b%5B2%5D%29%2Cgreen%28c%5B2%5D%29%2Cgreen%28a%5B2%5D%29%2Cgreen%28b%5B2%5D%29%2Ca%5B3%5D%2Cb%5B3%5D%2Cc%5B3%5D%2Ca%5B3%5D%2Cb%5B3%5D%29%29=red%28a%5B1%5D%29%2Agreen%28b%5B2%5D%29%2Ac%5B3%5D%2Bred%28b%5B1%5D%29%2Agreen%28c%5B2%5D%29%2Aa%5B3%5D%2Bred%28c%5B1%5D%29%2Agreen%28a%5B2%5D%29%2Ab%5B3%5D-a%5B3%5D%2Agreen%28b%5B2%5D%29%2Ared%28c%5B1%5D%29-b%5B3%5D%2Agreen%28c%5B2%5D%29%2Ared%28a%5B1%5D%29-c%5B3%5D%2Agreen%28a%5B2%5D%29%2Ared%28b%5B1%5D%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "D for this problem is:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D=%28matrix%283%2C5%2C2%2C1%2C-5%2C2%2C1%2C1%2C2%2C-2%2C1%2C2%2C1%2C-4%2C1%2C1%2C-4%29%29=2%2A2%2A1%2B1%2A%28-2%29%2A1%2B%28-5%29%2A1%2A%28-4%29-1%2A2%2A%28-5%29-%28-4%29%2A%28-2%29%2A2-1%2A1%2A1=4-2%2B20%2B10-16-1=25 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "And\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D%5Bx%5D=%28matrix%283%2C5%2C-21%2C1%2C-5%2C-21%2C1%2C-15%2C2%2C-2%2C-15%2C2%2C18%2C-4%2C1%2C18%2C-4%29%29=-42-36-300%2B180%2B168%2B15=-25 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "And finally,\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D%5Bx%5D%2FD=%28-25%29%2F25=-1 ALIGN=MIDDLE  ALT=\"D%5Bx%5D%2FD=%28-25%29%2F25=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, just as we suspected.  Now that you know how to do it, I'll let you handle the arithmetic for the other two variables.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Hope that helps,<BR>\n" );
document.write( "John </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );