document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=593730>Question 593730</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I am just re-learning algebra after 20 years, so it's been difficult.\r<BR>\n" );
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document.write( "A college student earned $7300 during summer vacation working as a waiter in a popular restaurant.  The student invested part of the money at 7% and the rest at 6%.  If the student received a total of $458 in interest at the end of the yera, how much was invested at 7%?\r<BR>\n" );
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document.write( "Thank you. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #376435 by </B><A HREF=/tutors/aboutme.mpl?userid=bucky><B>bucky(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=bucky><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=376435\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> You have two unknowns, the amount of money invested at 7% (let's call that amount X) and the amount of money invested at 6% (let's call that amount Y). With two unknowns you will need two equations to find the two amounts.<BR>\n" );
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document.write( "Since X plus Y equals the total money invested (which the problem says is $7300) we can write one equation (the first) as:<BR>\n" );
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document.write( "X + Y = 7300<BR>\n" );
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document.write( "Next, we can see that when the amount X is invested at 7% the amount of interest it will make in a year can be found by multiplying X times 0.07 (the decimal equivalent of 7%). Similarly when the amount Y is invested at 6% the amount of interest it will make in a year can be found by multiplying Y times 0.06. The problem tells you that if you add these two amounts of interest, the total interest will be $458. We can, therefore, write the equation for this. Our second equation is:<BR>\n" );
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document.write( "0.07X + 0.06Y = 458<BR>\n" );
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document.write( "We can, for convenience, get rid of the decimals in this second equation by multiplying both sides of the equation (all terms) by 100. In doing that we convert the equation to:<BR>\n" );
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document.write( "7X + 6Y = 45800<BR>\n" );
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document.write( "In summary, our two equations are:<BR>\n" );
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document.write( "X + Y = 7300 and<BR>\n" );
document.write( "7X + 6Y = 45800<BR>\n" );
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document.write( "There are several ways we could solve this system of two equations. One of the ways would be to solve our first equation for Y in terms of X by subtracting X from both sides to get Y = 7300 - X. Then we could substitute 7300 - X for its equivalent value of Y in the second equation. This makes the second equation become:<BR>\n" );
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document.write( "7X + 6(7300 - X) = 45800<BR>\n" );
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document.write( "Do the distributed multiplication by multiplying 6 times each of the terms in the parentheses to make the equation become:<BR>\n" );
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document.write( "7X + 43800 - 6X = 45800<BR>\n" );
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document.write( "Combine the terms involving X by subtracting the 6X from the 7X to get just X which simplifies the equation to:<BR>\n" );
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document.write( "X + 43800 = 45800<BR>\n" );
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document.write( "Finally solve for X by subtracting the 43800 from both sides to get:<BR>\n" );
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document.write( "X = 2000<BR>\n" );
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document.write( "This says that X (the amount invested at 7% is equal to $2,000). That means that the remainder of the $7300 ... namely $5,300 ... is invested at 6%.<BR>\n" );
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document.write( "Check by multiplying 7% times 2000 and 6% by 5300 to see if they total $458 dollars. 7% times $2000 is $140 and 6% times $5300 is $318. Add the two together to find that $140 + $318 = $458, just as the problem said it should. <BR>\n" );
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document.write( "Another way you could have done it is to start with the two equations:<BR>\n" );
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document.write( "X + Y = 7300 and<BR>\n" );
document.write( "7X + 6Y = 45800<BR>\n" );
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document.write( "Let's multiply the top equation by 6 (all terms on both sides) to make the top equation become 6X + 6Y = 43800. This with the second equation can be written as:<BR>\n" );
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document.write( "6X + 6Y = 43800 and<BR>\n" );
document.write( "7X + 6Y = 45800<BR>\n" );
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document.write( "Subtract these two equations in vertical columns. Note that the 6X minus 6X in the second column equals zero and therefore disappears.  The other subtractions result in:<BR>\n" );
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document.write( "-X = -2000<BR>\n" );
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document.write( "and multiply both sides by -1 to convert this equation to:<BR>\n" );
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document.write( "X = 2000<BR>\n" );
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document.write( "This is the same answer we got previously. It's just another way of doing it.<BR>\n" );
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document.write( "Hope this helps you to understand the problem a little better. Good luck with your efforts to recall algebra. Use this site to help you. Maybe we'll hook up again.<BR>\n" );
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