document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1152425>Question 1152425</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Frank bought 2 shirts and 1 pair of dress pants for a total of $55. If he had bought 1 shirt and 2 pairs of dress pants, he would have paid $68. What was the price of each shirt and each pair of dress pants? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #774457 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=774457\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "Translating the given information directly into equations gives us<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%2By+=+55 ALIGN=MIDDLE  ALT=\"2x%2By+=+55\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B2y+=+68 ALIGN=MIDDLE  ALT=\"x%2B2y+=+68\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "While solving that pair of equations using substitution, as the other tutor did, is valid, there are other ways that are far less work.<br><BR>\n" );
document.write( "The solution with formal algebra using substitution is hard to interpret in everyday language; there are at least a couple of ways to solve the pair of equations using formal algebra for which it is easy to see how common sense makes the solution processes easy to understand.<br><BR>\n" );
document.write( "Here are a couple of solutions that use algebra that is easily described in everyday language.<br><BR>\n" );
document.write( "(1) One standard method for solving pairs of linear equations -- and the most straightforward when the information is given in this way -- is elimination: multiply one or both equations by some constant so that you can use the two resulting equations to eliminate one variable by either adding or subtracting the two equations.<br><BR>\n" );
document.write( "In common sense language, suppose that, instead of buying 1 shirt and 2 pairs of pants for $68, he bought 2 shirts and 4 pairs of pants.  That would cost him $136.  Algebraically, \"multiply the second equation by 2\":<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B2y+=+68 ALIGN=MIDDLE  ALT=\"x%2B2y+=+68\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%2B4y+=+136 ALIGN=MIDDLE  ALT=\"2x%2B4y+=+136\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Now compare the two purchases (subtract the first equation from this new equation):<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%2B4y+=+136 ALIGN=MIDDLE  ALT=\"2x%2B4y+=+136\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%2By+=+55 ALIGN=MIDDLE  ALT=\"2x%2By+=+55\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3y+=+81 ALIGN=MIDDLE  ALT=\"3y+=+81\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> y=27<br><BR>\n" );
document.write( "In everyday language, the difference between the two purchases is now 3 extra pairs of pants, for an extra $81; that means each pair of pants costs $81/3 = $27.<br><BR>\n" );
document.write( "Then use that fact with any of the purchases to determine the cost of each shirt.<br><BR>\n" );
document.write( "(2) And here is a method that uses an unusual algebraic approach but is very understandable with common sense.<br><BR>\n" );
document.write( "Between the two purchases described in the problem, the difference is one more pair of pants and one less shirt, for an increase of $13 in the total cost.  (Algebraically, to find the difference between the two purchases, subtract the first equation from the second):<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%2By+=+55 ALIGN=MIDDLE  ALT=\"2x%2By+=+55\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B2y+=+68 ALIGN=MIDDLE  ALT=\"x%2B2y+=+68\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-x+=+13 ALIGN=MIDDLE  ALT=\"y-x+=+13\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "So let's make a third purchase where we again add one pair of pants and subtract one shirt.  That gives us three pairs of pants and no shirts, for a cost of $68+$13 = $81.<br><BR>\n" );
document.write( "So again we find the cost of each pair of pants to be $81/3 = $27. </SPAN>\n" );
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