document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=499319>Question 499319</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help me understand Cramer's rule.  My system is 3x+2y=6;2x+2y=8. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #337564 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=337564\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <TABLE  BORDER=1>\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=cramers_rule_2x2.solver>Using Cramer's Rule to Solve Systems with 2 variables</A></TD></TR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%283%2Ax%2B2%2Ay=6%2C2%2Ax%2B2%2Ay=8%29 ALIGN=MIDDLE  ALT=\"system%283%2Ax%2B2%2Ay=6%2C2%2Ax%2B2%2Ay=8%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  First let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=%28matrix%282%2C2%2C3%2C2%2C2%2C2%29%29 ALIGN=MIDDLE  ALT=\"A=%28matrix%282%2C2%2C3%2C2%2C2%2C2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. This is the matrix formed by the coefficients of the given system of equations.<BR>\n" );
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document.write( "  Take note that the right hand values of the system are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6 ALIGN=MIDDLE  ALT=\"6\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8 ALIGN=MIDDLE  ALT=\"8\"   DISABLED_style= \"max-width:90%; height:auto;\" > which are highlighted here: <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%283%2Ax%2B2%2Ay=highlight%286%29%2C2%2Ax%2B2%2Ay=highlight%288%29%29 ALIGN=MIDDLE  ALT=\"system%283%2Ax%2B2%2Ay=highlight%286%29%2C2%2Ax%2B2%2Ay=highlight%288%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  These values are important as they will be used to replace the columns of the matrix A.<BR>\n" );
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document.write( "  Now let's calculate the the <a  rel=nofollow HREF=http://en.wikipedia.org/wiki/Determinant>determinant</a> of the matrix A to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28A%29=%283%29%282%29-%282%29%282%29=2 ALIGN=MIDDLE  ALT=\"abs%28A%29=%283%29%282%29-%282%29%282%29=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Remember that the determinant of the 2x2 matrix <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=%28matrix%282%2C2%2Ca%2Cb%2Cc%2Cd%29%29 ALIGN=MIDDLE  ALT=\"A=%28matrix%282%2C2%2Ca%2Cb%2Cc%2Cd%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28A%29=ad-bc ALIGN=MIDDLE  ALT=\"abs%28A%29=ad-bc\"   DISABLED_style= \"max-width:90%; height:auto;\" >. If you need help with calculating the determinant of any two by two matrices, then check out this <a  rel=nofollow HREF=http://www.algebra.com/algebra/homework/Matrices-and-determiminant/determinant-of-2x2-matrix.solver>solver</a>.<BR>\n" );
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document.write( "  Notation note: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28A%29 ALIGN=MIDDLE  ALT=\"abs%28A%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > denotes the determinant of the matrix A.<BR>\n" );
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document.write( "  Now replace the first column of A (that corresponds to the variable 'x') with the values that form the right hand side of the system of equations. We will denote this new matrix <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5Bx%5D ALIGN=MIDDLE  ALT=\"A%5Bx%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > (since we're replacing the 'x' column so to speak).<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5Bx%5D=%28matrix%282%2C2%2Chighlight%286%29%2C2%2Chighlight%288%29%2C2%29%29 ALIGN=MIDDLE  ALT=\"A%5Bx%5D=%28matrix%282%2C2%2Chighlight%286%29%2C2%2Chighlight%288%29%2C2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  Now compute the <a  rel=nofollow HREF=http://www.algebra.com/algebra/homework/Matrices-and-determiminant/determinant-of-2x2-matrix.solver>determinant</a> of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5Bx%5D ALIGN=MIDDLE  ALT=\"A%5Bx%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28A%5Bx%5D%29=%286%29%282%29-%282%29%288%29=-4 ALIGN=MIDDLE  ALT=\"abs%28A%5Bx%5D%29=%286%29%282%29-%282%29%288%29=-4\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Once again, remember that the determinant of the 2x2 matrix <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=%28matrix%282%2C2%2Ca%2Cb%2Cc%2Cd%29%29 ALIGN=MIDDLE  ALT=\"A=%28matrix%282%2C2%2Ca%2Cb%2Cc%2Cd%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28A%29=ad-bc ALIGN=MIDDLE  ALT=\"abs%28A%29=ad-bc\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  To find the first solution, simply divide the determinant of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5Bx%5D ALIGN=MIDDLE  ALT=\"A%5Bx%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > by the determinant of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A ALIGN=MIDDLE  ALT=\"A\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%28abs%28A%5Bx%5D%29%29%2F%28abs%28A%29%29=%28-4%29%2F%282%29=-2 ALIGN=MIDDLE  ALT=\"x=%28abs%28A%5Bx%5D%29%29%2F%28abs%28A%29%29=%28-4%29%2F%282%29=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  So the first solution is <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;\" ><BR>\n" );
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document.write( "  We'll follow the same basic idea to find the other solution. Let's reset by letting <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=%28matrix%282%2C2%2C3%2C2%2C2%2C2%29%29 ALIGN=MIDDLE  ALT=\"A=%28matrix%282%2C2%2C3%2C2%2C2%2C2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > again (this is the coefficient matrix).<BR>\n" );
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document.write( "  Now replace the second column of A (that corresponds to the variable 'y') with the values that form the right hand side of the system of equations. We will denote this new matrix <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5By%5D ALIGN=MIDDLE  ALT=\"A%5By%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > (since we're replacing the 'y' column in a way).<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5Bx%5D=%28matrix%282%2C2%2C3%2Chighlight%286%29%2C2%2Chighlight%288%29%29%29 ALIGN=MIDDLE  ALT=\"A%5Bx%5D=%28matrix%282%2C2%2C3%2Chighlight%286%29%2C2%2Chighlight%288%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  Now compute the <a  rel=nofollow HREF=http://www.algebra.com/algebra/homework/Matrices-and-determiminant/determinant-of-2x2-matrix.solver>determinant</a> of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5By%5D ALIGN=MIDDLE  ALT=\"A%5By%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28A%5By%5D%29=%283%29%288%29-%286%29%282%29=12 ALIGN=MIDDLE  ALT=\"abs%28A%5By%5D%29=%283%29%288%29-%286%29%282%29=12\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
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document.write( "  To find the second solution, divide the determinant of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5By%5D ALIGN=MIDDLE  ALT=\"A%5By%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > by the determinant of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A ALIGN=MIDDLE  ALT=\"A\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28abs%28A%5By%5D%29%29%2F%28abs%28A%29%29=%2812%29%2F%282%29=6 ALIGN=MIDDLE  ALT=\"y=%28abs%28A%5By%5D%29%29%2F%28abs%28A%29%29=%2812%29%2F%282%29=6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  So the second solution is <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;\" ><BR>\n" );
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document.write( "  Final Answer:<BR>\n" );
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document.write( "  So the solutions are <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;\" > and <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;\" > giving the ordered pair (-2, 6)<BR>\n" );
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document.write( "  Once again, Cramer's Rule is dependent on determinants. Take a look at this <a  rel=nofollow HREF=http://www.algebra.com/algebra/homework/Matrices-and-determiminant/determinant-of-2x2-matrix.solver>2x2 Determinant Solver</a> if you need more practice with determinants.<BR>\n" );
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