document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=662667>Question 662667</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Ivan is three years older than his sister Mary. One-third of Ivan's age is two years less than one-half his sister's age one year ago. Find their age's now. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #412439 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=412439\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's write this algebraically. <BR>\n" );
document.write( "Ivan (I) is three years older (+3) than his sister Mary (M). [M+3=I]<BR>\n" );
document.write( "One third (1/3) of Ivan's (I) age is 2 years less (-2) than one-half his sister's age (1/2M) one year ago (-1). [1/3I=1/2M-3]<BR>\n" );
document.write( "We can now describe this as a system. Let's assume I is X and M is Y. Let's put our equations into x+y=# form.<BR>\n" );
document.write( "<P>M+3=I<BR>\n" );
document.write( "M=I+3<BR>\n" );
document.write( "M-I=3<BR>\n" );
document.write( "-X+Y=3\r<BR>\n" );
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document.write( "<p>1/3I=1/2M+3<BR>\n" );
document.write( "2/3I=M+3<BR>\n" );
document.write( "2/3I-M=3<BR>\n" );
document.write( "2/3X-Y=3\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( "  First let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=%28matrix%282%2C2%2C-1%2C1%2C0.666666666666667%2C-1%29%29 ALIGN=MIDDLE  ALT=\"A=%28matrix%282%2C2%2C-1%2C1%2C0.666666666666667%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=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   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%28-1%2Ax%2B1%2Ay=highlight%283%29%2C0.666666666666667%2Ax%2B-1%2Ay=highlight%283%29%29 ALIGN=MIDDLE   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=%28-1%29%28-1%29-%281%29%280.666666666666667%29=0.333333333333333 ALIGN=MIDDLE  ALT=\"abs%28A%29=%28-1%29%28-1%29-%281%29%280.666666666666667%29=0.333333333333333\"   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%283%29%2C1%2Chighlight%283%29%2C-1%29%29 ALIGN=MIDDLE  ALT=\"A%5Bx%5D=%28matrix%282%2C2%2Chighlight%283%29%2C1%2Chighlight%283%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=%283%29%28-1%29-%281%29%283%29=-6 ALIGN=MIDDLE  ALT=\"abs%28A%5Bx%5D%29=%283%29%28-1%29-%281%29%283%29=-6\"   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=%28-6%29%2F%280.333333333333333%29=-18 ALIGN=MIDDLE  ALT=\"x=%28abs%28A%5Bx%5D%29%29%2F%28abs%28A%29%29=%28-6%29%2F%280.333333333333333%29=-18\"   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=-18 ALIGN=MIDDLE  ALT=\"x=-18\"   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%2C-1%2C1%2C0.666666666666667%2C-1%29%29 ALIGN=MIDDLE  ALT=\"A=%28matrix%282%2C2%2C-1%2C1%2C0.666666666666667%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%2C-1%2Chighlight%283%29%2C0.666666666666667%2Chighlight%283%29%29%29 ALIGN=MIDDLE   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=%28-1%29%283%29-%283%29%280.666666666666667%29=-5 ALIGN=MIDDLE  ALT=\"abs%28A%5By%5D%29=%28-1%29%283%29-%283%29%280.666666666666667%29=-5\"   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=%28-5%29%2F%280.333333333333333%29=-15 ALIGN=MIDDLE  ALT=\"y=%28abs%28A%5By%5D%29%29%2F%28abs%28A%29%29=%28-5%29%2F%280.333333333333333%29=-15\"   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=-15 ALIGN=MIDDLE  ALT=\"y=-15\"   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=-18 ALIGN=MIDDLE  ALT=\"x=-18\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-15 ALIGN=MIDDLE  ALT=\"y=-15\"   DISABLED_style= \"max-width:90%; height:auto;\" > giving the ordered pair (-18, -15)<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>There's no such thing as a negative age, so just flip that. That means <b>Ivan is 18 and Mary is 15.</b><BR>\n" );
document.write( "<p>Hope this helps! </SPAN>\n" );
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