document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=206918>Question 206918</A>: <I><SPAN STYLE=\"word-break: break-all;\"> This is not for homework; however, it is one in a homework text for the University of Phoenix Introductory and Intermediate Algebra.\r<BR>\n" );
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document.write( "In order to disuss with my classmates about how to solve linear equations, I need to be able to understand this one by today.  Rod is a pilot for Crossland Airways. He computes his flight time against a headwind for a tip of 2900 mi. at 5 hr. The flight would take 4 hr and 50 min if the headwind were half as great.  Find the headwind and the plane's air speed.\r<BR>\n" );
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document.write( "I understand I must use d=rxt. How do I come up with the two equations?  I thought maybe it was 2,900= R x 5  and D = r 1/2 x 4.50; but, they do not look right.  I came up with an air speed of 124.7?  Please help.Reference:\r<BR>\n" );
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document.write( " Bittenger, M. L. & Beecher, J. A. (2007). Introductory and intermediate algebra (3rd ed.). Boston: Pearson-Addison Wesley.\r<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 #156475 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=156475\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Rod is a pilot for Crossland Airways. He computes his flight time against a headwind for a trip of 2900 mi. at 5 hr. The flight would take 4 hr and 50 min if the headwind were half as great.  Find the headwind and the plane's air speed.<BR>\n" );
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document.write( "let x = rate of speed of airplane.<BR>\n" );
document.write( "let y = rate of speed of headwind<BR>\n" );
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document.write( "let D = distance = 2900 miles<BR>\n" );
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document.write( "let T = time in minutes<BR>\n" );
document.write( "5 hours = 5 hours = 300 minutes<BR>\n" );
document.write( "4 hours 50 minutes = 290 minutes<BR>\n" );
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document.write( "converting to minutes should make the arithmetic a little easier even though the number will get large.<BR>\n" );
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document.write( "formula is R * T = D where R is the combined rate of speed and T is the time it takes in minutes and D is the distance in miles.<BR>\n" );
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document.write( "let R = (x-y)<BR>\n" );
document.write( "let T = 300 minutes<BR>\n" );
document.write( "formula becomes:<BR>\n" );
document.write( "(x - y) * 300 = 2900<BR>\n" );
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document.write( "if the rate of the headwind is half as great, then the rate of the headwind becomes y/2<BR>\n" );
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document.write( "let R = (x - (y/2))<BR>\n" );
document.write( "Let T = 290 minutes<BR>\n" );
document.write( "formula becomes:<BR>\n" );
document.write( "(x - (y/2)) * 290 = 2900<BR>\n" );
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document.write( "we have 2 formulas to solve simultaneously.<BR>\n" );
document.write( "(x - y) * 300 = 2900<BR>\n" );
document.write( "(x - (y/2)) * 290 = 2900<BR>\n" );
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document.write( "remove parentheses and simplify to get:<BR>\n" );
document.write( "300*x - 300*y = 2900<BR>\n" );
document.write( "290*x - 145*y = 2900<BR>\n" );
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document.write( "you can solve these in several ways.<BR>\n" );
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document.write( "solve by substitution means you use one of the equations to get x in terms of y or y in terms of x and then substitute in the other equation.<BR>\n" );
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document.write( "solve by elimination means you multiply both sides of the equations by factors that will allow one of the unknown variables to cancel out once you add the equations together.<BR>\n" );
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document.write( "since both equations equal the same value, you can set each equation equal to each other.<BR>\n" );
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document.write( "we'll solve by elimination.<BR>\n" );
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document.write( "your equations after simplification are:<BR>\n" );
document.write( "300*x - 300*y = 2900<BR>\n" );
document.write( "290*x - 145*y = 2900<BR>\n" );
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document.write( "if you multiply the first equation by 290 and you multiply the second equation by 300 then you should be able to eliminate the x when you add or subtract both equations from each other.<BR>\n" );
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document.write( "after multiplication, the equations become:<BR>\n" );
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document.write( "87000*x - 87000*y = 841000<BR>\n" );
document.write( "87000*x - 43500*y = 870000<BR>\n" );
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document.write( "subtract the second equation from the first equation to get:<BR>\n" );
document.write( "- 43500 * y = - 29000<BR>\n" );
document.write( "divide both sides of this equation by (-43500) to get y = .667 = 2/3 mile per minute.<BR>\n" );
document.write( "if y = 2/3, then y/2 = 1/3 mile per minute.<BR>\n" );
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document.write( "now that we know the rate of speed of the headwind, we can solve for the rate of speed of the airplane<BR>\n" );
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document.write( "in our first equation, we had:<BR>\n" );
document.write( "300*x - 300*y = 2900<BR>\n" );
document.write( "this equation becomes:<BR>\n" );
document.write( "300*x - 300*(2/3) = 2900 which becomes:<BR>\n" );
document.write( "300*x - 200 = 2900 which becomes:<BR>\n" );
document.write( "300*x = 3100 which becomes:<BR>\n" );
document.write( "x = 10 and 1/3 equals 31/3 miles per minute.<BR>\n" );
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document.write( "in our second equation, we now have:<BR>\n" );
document.write( "290*(31/3) - 290*(1/3) = 2900 which becomes:<BR>\n" );
document.write( "8990/3 - 290/3 = 2900 which becomes:<BR>\n" );
document.write( "8700/3 = 2900 which becomes:<BR>\n" );
document.write( "2900 = 2900 which means that the value of 31/3 miles per minute for the plane and the value of 2/3 miles per minute for the headwind is accurate.<BR>\n" );
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document.write( "converting these back to hours we get:<BR>\n" );
document.write( "speed of the plane is 60*31/3 = 620 miles per hour.<BR>\n" );
document.write( "speed of the headwind is 60*2/3 = 40 miles per hour.<BR>\n" );
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document.write( "you can plug these values into the original equations to see that they work as well.<BR>\n" );
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