document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=611044>Question 611044</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Ten years from now, A will be twice as old as B. Five years ago, A was three times as old as B.  What are the present ages of A and B?<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 #384746 by </B><A HREF=/tutors/aboutme.mpl?userid=radh><B>radh(108)</B></A><img src=\"/images/stars/stars-09.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=radh><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=radh><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=384746\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's map this out mathematically<BR>\n" );
document.write( "Then years from now, (+10) A will be twice as old as B (2b). Five years ago, (-5) A was three times as old as B (3b).\r<BR>\n" );
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document.write( "<p>So, that means:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2b%2B10=a ALIGN=MIDDLE  ALT=\"2b%2B10=a\"   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=3b-5=a ALIGN=MIDDLE  ALT=\"3b-5=a\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "<p>Let's simplify that and move both variables on one side of the equation to get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3b-a=5 ALIGN=MIDDLE  ALT=\"3b-a=5\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<p>We're going to substitute this into the solver. x=b and y=a. \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=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%282%2Ax%2B-1%2Ay=-10%2C3%2Ax%2B-1%2Ay=5%29 ALIGN=MIDDLE  ALT=\"system%282%2Ax%2B-1%2Ay=-10%2C3%2Ax%2B-1%2Ay=5%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%2C2%2C-1%2C3%2C-1%29%29 ALIGN=MIDDLE  ALT=\"A=%28matrix%282%2C2%2C2%2C-1%2C3%2C-1%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=-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=5 ALIGN=MIDDLE  ALT=\"5\"   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%282%2Ax%2B-1%2Ay=highlight%28-10%29%2C3%2Ax%2B-1%2Ay=highlight%285%29%29 ALIGN=MIDDLE  ALT=\"system%282%2Ax%2B-1%2Ay=highlight%28-10%29%2C3%2Ax%2B-1%2Ay=highlight%285%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=%282%29%28-1%29-%28-1%29%283%29=1 ALIGN=MIDDLE  ALT=\"abs%28A%29=%282%29%28-1%29-%28-1%29%283%29=1\"   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%28-10%29%2C-1%2Chighlight%285%29%2C-1%29%29 ALIGN=MIDDLE  ALT=\"A%5Bx%5D=%28matrix%282%2C2%2Chighlight%28-10%29%2C-1%2Chighlight%285%29%2C-1%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=%28-10%29%28-1%29-%28-1%29%285%29=15 ALIGN=MIDDLE  ALT=\"abs%28A%5Bx%5D%29=%28-10%29%28-1%29-%28-1%29%285%29=15\"   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=%2815%29%2F%281%29=15 ALIGN=MIDDLE  ALT=\"x=%28abs%28A%5Bx%5D%29%29%2F%28abs%28A%29%29=%2815%29%2F%281%29=15\"   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=15 ALIGN=MIDDLE  ALT=\"x=15\"   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%2C2%2C-1%2C3%2C-1%29%29 ALIGN=MIDDLE  ALT=\"A=%28matrix%282%2C2%2C2%2C-1%2C3%2C-1%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%2C2%2Chighlight%28-10%29%2C3%2Chighlight%285%29%29%29 ALIGN=MIDDLE  ALT=\"A%5Bx%5D=%28matrix%282%2C2%2C2%2Chighlight%28-10%29%2C3%2Chighlight%285%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=%282%29%285%29-%28-10%29%283%29=40 ALIGN=MIDDLE  ALT=\"abs%28A%5By%5D%29=%282%29%285%29-%28-10%29%283%29=40\"   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=%2840%29%2F%281%29=40 ALIGN=MIDDLE  ALT=\"y=%28abs%28A%5By%5D%29%29%2F%28abs%28A%29%29=%2840%29%2F%281%29=40\"   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=40 ALIGN=MIDDLE  ALT=\"y=40\"   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=15 ALIGN=MIDDLE  ALT=\"x=15\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=40 ALIGN=MIDDLE  ALT=\"y=40\"   DISABLED_style= \"max-width:90%; height:auto;\" > giving the ordered pair (15, 40)<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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document.write( "<p><b>So, person A is 40 and person B is 15.</b> :) </SPAN>\n" );
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