document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1063361>Question 1063361</A>: <I><SPAN STYLE=\"word-break: break-all;\"> My book doesn't show me how to do this. Any steps and explanations would be great.\r<BR>\n" );
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document.write( "Assume that a,b, and c are integers and a is not equal to 0.<BR>\n" );
document.write( "a. Proof Prove that the solution of the linear equation ax-b=c must be a rational number.<BR>\n" );
document.write( "b. Writing Describe the values of a, b, and c for which the solutions of ax^2+b=c are rational. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #678460 by </B><A HREF=/tutors/aboutme.mpl?userid=rothauserc><B>rothauserc(4718)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=rothauserc><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=rothauserc><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=678460\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The set of rational numbers with the operations of addition, subtraction, multiplication and division defined, this structure is called a Field(F)<BR>\n" );
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document.write( "We are given a, b, c are integers and a is not equal to 0, therefore a, b, c are rational numbers<BR>\n" );
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document.write( "Note that a = a / 1, b = b / 1 and c = c / 1<BR>\n" );
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document.write( "a.  ax - b = c<BR>\n" );
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document.write( "add b to both sides of =, b + (-b) = 0 since -b is the additive inverse of b<BR>\n" );
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document.write( "ax = c + b<BR>\n" );
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document.write( "since a is not = 0 there exists a^(-1) which is the multiplicative inverse of a, that is a * a^(-1) = 1 where * is multiplication<BR>\n" );
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document.write( "x = (c + b) * a^(-1)<BR>\n" );
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document.write( "a^(-1) = 1/a<BR>\n" );
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document.write( "x = (c + b) / a<BR>\n" );
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document.write( "Note that F is closed under addition, this means that given c and b are rational, then c + b = d is rational<BR>\n" );
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document.write( "then d / a is rational(can be expressed as a fraction)<BR>\n" );
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document.write( "b. ax^2 + b = c<BR>\n" );
document.write( "ax^2 = (c - b)<BR>\n" );
document.write( "x^2 = (c - b) / a<BR>\n" );
document.write( "x = sqrt( (c - b) / a )<BR>\n" );
document.write( "x is rational if a not = 0, b , c are rational numbers with c > or = b<BR>\n" );
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document.write( "Note that 0 is a rational number<BR>\n" );
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document.write( " </SPAN>\n" );
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