document.write( "Question 1033597: if there are at least one of each type of coin, and there are 28 coins in total, what possible combinations of nickles, dimes, and quarters will add up to 2 dollars using matricies \n" ); document.write( "

Algebra.Com's Answer #649046 by robertb(5830) $\"\"$ $\"About$

You can put this solution on YOUR website!
Let n = number of nickels, d = number of dimes, and q = number of quarters.\r
\n" ); document.write( "\n" ); document.write( "We have to solve the system\r
\n" ); document.write( "\n" ); document.write( "n+d+q = 28
\n" ); document.write( "0.05n+0.10d+0.25q = 2.00\r
\n" ); document.write( "\n" ); document.write( "The second equation is equivalent to n + 2d + 5q = 40, after simplification.\r
\n" ); document.write( "\n" ); document.write( "The system represented as an augmented matrix is \r
\n" ); document.write( "\n" ); document.write( " $\"%28matrix%282%2C4%2C1%2C1%2C1%2C28%2C1%2C2%2C5%2C40%29%29\"$
\n" ); document.write( "~ $\"%28matrix%282%2C4%2C1%2C1%2C1%2C28%2C0%2C1%2C4%2C12%29%29\"$ ~ $\"%28matrix%282%2C4%2C1%2C0%2C-3%2C16%2C0%2C1%2C4%2C12%29%29\"$ \r
\n" ); document.write( "\n" ); document.write( "==> n-3q = 16 and d+4q = 12.\r
\n" ); document.write( "\n" ); document.write( "==> n = 3q+16 and d = 12 - 4q.\r
\n" ); document.write( "\n" ); document.write( "==> The possible values of q are 0,1, 2, and 3.
\n" ); document.write( "The possible triples (n,d,q) are thus
\n" ); document.write( "(16,12,0), (19,8,1), (22,4,2), and (25,0,3).\r
\n" ); document.write( "\n" ); document.write( "But since there is at least one of each type of coin, the only possible combinations are (19,8,1) and (22,4,2). \n" ); document.write( "

\n" );