document.write( "Question 1179804: in a block of flats there are 24 units of 3 types: the luxury unit, the superior unit and deluxe unit. the luxury can accommodate 8 people, the superior unit can accommodate 7 people and deluxe can accommodate 5 people. given that the total number of people living in this block is 160. How many of each type are there? \n" ); document.write( "

Algebra.Com's Answer #809405 by greenestamps(13200) $\"\"$ $\"About$

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\n" ); document.write( "x = # of luxury units
\n" ); document.write( "y = # of superior units
\n" ); document.write( "z = # of deluxe units

\n" ); document.write( " $\"x%2By%2Bz=24\"$ the total number of units is 24
\n" ); document.write( " $\"8x%2B7y%2B5z=160\"$ the total number of residents is 160

\n" ); document.write( "This is two equations with three variables, in which the values of the variables are non-negative integers. That is a system of Diophantine equations; there are standard processes for solving them.

\n" ); document.write( "We first need to eliminate one variable to get a single equation with two variables. Let's eliminate z using elimination.

\n" ); document.write( " $\"5x%2B5y%2B5z=120\"$
\n" ); document.write( " $\"8x%2B7y%2B5z=160\"$
\n" ); document.write( " $\"3x%2B2y=40\"$

\n" ); document.write( "Next we solve this single equation for one variable and use the requirement that the solutions be non-negative integers to find the solution(s) to the problem.

\n" ); document.write( " $\"y=%2840-3x%29%2F2\"$

\n" ); document.write( "We could at this point choose non-negative integers values for x and find the ones that give non-negative integer values for y. But we can make finding the solutions easier by going one step further with our algebra before we do that.

\n" ); document.write( "Perform the indicated division, expressing the result as a quotient and a remainder:

\n" ); document.write( " $\"y=%2840-3x%29%2F2+=+%2840-2x-x%29%2F2+=+%2820-x%29-x%2F2\"$

\n" ); document.write( "From this we see that y has to be an integer and 20-x is an integer, so x/2 has to be an integer. That tells us x has to be even.

\n" ); document.write( "Now we are ready to find the solutions to the problem.

\r\n" );
document.write( "  x  y=(20-x)-x/2 z=24-(x+y)\r\n" );
document.write( " ----------------------------\r\n" );
document.write( "  0   20-0=20      24-20=4\r\n" );
document.write( "  2   18-1=17      24-19=5\r\n" );
document.write( "  4   16-2=14      24-18=6\r\n" );
document.write( "  6   14-3=11      24-17=7\r\n" );
document.write( "  8   12-4=8       24-16=8\r\n" );
document.write( " 10   10-5=5       24-15=9\r\n" );
document.write( " 12    8-6=2       24-14=10

\n" ); document.write( "Larger values for x would produce negative values for y, so the list of solutions is complete.

\n" ); document.write( " \n" ); document.write( "

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