document.write( "Question 1015841: Find the value of m for which the pair of simultaneous equations 3x + my = 5 and (m + 2)x + 5y = m have:
\n" ); document.write( "a) infinitely many solutions
\n" ); document.write( "b) no solutions. \n" ); document.write( "

Algebra.Com's Answer #632262 by Fombitz(32388) $\"\"$ $\"About$

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Let's look at the coefficient matrix,
\n" ); document.write( " $\"A=%28matrix%282%2C2%2C%0D%0A3%2Cm%2C%0D%0Am%2B2%2C5%29%29\"$
\n" ); document.write( "The determinant is then,
\n" ); document.write( " $\"15-m%28m%2B2%29=15-m%5E2-2m=-m%5E2-2m%2B15\"$
\n" ); document.write( "Look for values of $\"m\"$ that make the determinant equal to zero,
\n" ); document.write( " $\"m%5E2-2m%2B15=0\"$
\n" ); document.write( " $\"%28m%2B5%29%28m-3%29=0\"$
\n" ); document.write( "So then when $\"m=-5\"$ , the equations become,
\n" ); document.write( " $\"3x-5y=5\"$
\n" ); document.write( " $\"-3x%2B5=-5\"$
\n" ); document.write( "and you see that the second equation is just the first equation multiplied by $\"-1\"$ .
\n" ); document.write( "So this dependent system has infinitely many solutions.
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\n" ); document.write( "When $\"m=3\"$ , the equations become,
\n" ); document.write( " $\"3x%2B3y=5\"$ or $\"x%2By=5%2F3\"$
\n" ); document.write( " $\"5x%2B5y=3\"$ or $\"x%2By=3%2F5\"$
\n" ); document.write( "So the equations are now parallel and therefore have no solution.
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