document.write( "Question 178699: for 100 crowns, a certain farmer bought some horses, goats, and sheep. For a horse, he paid 3 1/2 crowns, for a goat, 1 1/3 crowns, and for a sheep, 1/2 crown. Assuming he bought at least 7 of each and that he bought 100 animals in all, how many horses, goats, and sheep did he buy? \r
\n" ); document.write( "\n" ); document.write( "many thanks for your help. \n" ); document.write( "
Algebra.Com's Answer #133672 by ptaylor(2198)\"\" \"About 
You can put this solution on YOUR website!
Let x=number of horses
\n" ); document.write( "y=number of goats
\n" ); document.write( "z=number of sheep\r
\n" ); document.write( "\n" ); document.write( "Now we are told that:
\n" ); document.write( "x+y+z=100---------------------------eq1
\n" ); document.write( "and
\n" ); document.write( "(7/2)x+(4/3)y+(1/2)z=100 multiply each term by 6
\n" ); document.write( "21x+8y+3z=600---------------------------eq2
\n" ); document.write( "TWO EQUATIONS AND THREE UNKNOWNS USUALLY MEANS SOME TRIAL AND ERROR\r
\n" ); document.write( "\n" ); document.write( "multiply eq1 by 3 and subtract it form eq2 and we get:
\n" ); document.write( "18x+5y=300-----------------eq1a
\n" ); document.write( "Solve eq1a for y:
\n" ); document.write( "subtract 18x from each side
\n" ); document.write( "5y=300-18x divide each term by5
\n" ); document.write( "y=60-18x/5------------------------------eq1b\r
\n" ); document.write( "\n" ); document.write( "We know two very important things about this problem:
\n" ); document.write( "(1) We cannot have negative animals----we deal in positive numbers
\n" ); document.write( "(2) We cannot have fractions of animals---we deal in whole numbers
\n" ); document.write( "Now we are told that:
\n" ); document.write( "60-18x/5>=7 subtract 100 from each side
\n" ); document.write( "-18x/5>=7-60 or
\n" ); document.write( "-18x/5>=-53 bultiply each side by 5
\n" ); document.write( "-18x>=-265 divide each side by -18 (inequality sign changes)
\n" ); document.write( "x<= 14.7 or
\n" ); document.write( "x<=14 and we are told that x>=7, so
\n" ); document.write( "7<=x<=14
\n" ); document.write( "Now, by trial and error, we look at eq1b and determine which values of x between 7 and 14 (inclusive) will yield a whole number for y, starting with 7.
\n" ); document.write( "By inspection, we can see that in order for 18x to be divisible by 5, x must end in either a 0 or 5, so the only value for x that works between 7 and 14 is 10.
\n" ); document.write( "From eq1b:
\n" ); document.write( "If x=10, y=24
\n" ); document.write( "Substitute these values into eq1:
\n" ); document.write( "10+24+z=100
\n" ); document.write( "z=66
\n" ); document.write( "So, we have:
\n" ); document.write( "Horses=10
\n" ); document.write( "Goats=24
\n" ); document.write( "Sheep=66
\n" ); document.write( "Substitute these values into eq2 to make sure the money checks:
\n" ); document.write( "(21x+8y+3z=600)
\n" ); document.write( "21*10+8*24+3*66=600
\n" ); document.write( "210+192+198=600
\n" ); document.write( "600=600\r
\n" ); document.write( "\n" ); document.write( "Hope this helps---ptaylor\r
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