Algebra.Com's Answer #693611 by ikleyn(52906)![\"\"](\"/images/stars/stars-13.gif\")  
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document.write( "Find numbers a and b so that the system of equations
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document.write( "3x + y = 5
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document.write( "x - ay = b
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document.write( "has \r
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document.write( "(i) no solutions If a =  and b =/=  then the system HAS NO solutions.\r\n" );
document.write( " Indeed, the left sides of equations are proportional with the coefficient 3 (First to the Second),\r\n" );
document.write( " while the right sides are not proportional with the same coefficient.\r\n" );
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document.write( " The plot is two distinct parallel lines with NO intersection.\r\n" );
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document.write( "(ii) infinitely many solutions If a = and b = then the system HAS INFINITELY NANY solutions.\r\n" );
document.write( " Indeed, the left sides of equations are proportional with the coefficient 3 (First to the Second),\r\n" );
document.write( " while the right sides are not proportional with the same coefficient.\r\n" );
document.write( " Therefore, every solution to the first equation is the solution to the second equation.\r\n" );
document.write( " Thus the two equations are equivalent to one, either of the two.\r\n" );
document.write( " The plot is two coinsiding parallel lines with infinitely many common points . . .\r\n" );
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document.write( "(iii) a unique solution at (1,2). If a =/= then the system has a unique solution.\r\n" );
document.write( " The plot is two non-parallel straight lines that have a unique intersection, which represents \r\n" );
document.write( " the common solution to the system.\r\n" );
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document.write( "See the lesson\r
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document.write( " - Geometric interpretation of a linear system of two equations in two unknowns \r
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document.write( "in this site.\r
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document.write( "Also, you have this free of charge online textbook in ALGEBRA-I in this site\r
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document.write( " - ALGEBRA-I - YOUR ONLINE TEXTBOOK.\r
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document.write( "The referred lessons are the part of this online textbook under the topic \"Systems of two linear equations in two unknowns\".\r
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