document.write( "Question 827152: There are an infinite number of solutions to the system ax+2y=6k and 3x+by=15, so the sum of a and b must be? \n" ); document.write( "

Algebra.Com's Answer #498514 by jsmallt9(3758) $\"\"$ $\"About$

You can put this solution on YOUR website!
Are you sure the problem says \"sum\" of and b? If it does then don't think the problem can be solved (and you will have to re-post it). But if it says \"product\" or \"a*b\" then keep reading...

\n" ); document.write( "The quickest way to figure out what the product of a and b must be is to use determinants. The determinant of the coefficients:
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document.write( "|  a  2  |
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document.write( "|  3  b  |
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document.write( "... will be zero if there are an infinite number of solutions.

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document.write( "If you do not know about determinants then we can try solving the system:
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document.write( "ax + 2y = 6k
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document.write( "3x + by = 15
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document.write( "Multiplying both sides of the first equation by -3 and both sides of the second equation by \"a\", we get:
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document.write( "-3ax + -6y = -18k
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document.write( " 3ax + aby = 15a
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document.write( "The x terms cancel when we add the equations leaving:
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document.write( "-6y + aby = -18k + 15a
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document.write( "Factoring out y we get:
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document.write( "(-6 + ab)y = -18k + 15a
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document.write( "The x terms have already canceled out. When a system has an infinite number of solutions, then both the x's and the y's cancel out at the same time! So (-6 + ab) must be zero:
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document.write( "(-6 + ab) = 0
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document.write( "ab = 6
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document.write( "So there will be an infinite number of solutions if the product of a and b is 6. (You should get 6 if you used determinants, too.) \n" );
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