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word problems and turning them into algebraic equation to slove

Tara looks out into the park and sees pigs. chickens and ducks. She counts 30 heads and 80 feet. The sum of the number of pigs and chickens is 6 more than the number of ducks. ( all animals have normal amount ofheads and feet)
HOW MANY pigs, chickens and ducks are out there? Set up equation and solve
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2 Answers

Hi Nancy;
 
FIRST EQUATION...P+C+D=30
SECOND EQUATION...P+C=D+6
 
4P+2C+2D=80
Let's divide both sides by 2...
THIRD EQUATION...2P+C+D=40
 
Let's begin with the second equation..
P+C=D+6
This is also...
P+C-D=6
Let's add the first equation and second equation....
P+C-D=6
+(P+C+D=30)
2P+2C=36
Let's divide both sides by 2...
P+C=18--------we will use this again below...
The first equation was...
P+C+D=30
Let's substitute P+C with 18...
18+D=30
Let's subtract 18 from both sides...
D=12
Let's return to the third equation...
2P+C+D=40
2P+C+12=40
Let's subtract 12 from both sides...
2P+C=28
 
Let's subtract formulas...
2P+C=28
-(P+C=18)------from above
P=10
 
Let's return to the first equation...
P+C+D=30
10+C+12=30
C+22=30
C=8
 
Let's double check this with the third equation...
2P+C+D=40
2(10)+8+12=40
20+8+12=40
40=40
 

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Let set p for pigs, c for chickens, and d for ducks
 
(1)   p+c+d= 30  (She counts 30 heads, each animal has one head)
 
(2)   4p + 2c + 2d = 80 (each pig has 4 feet. each chicken and duck has 2 feet )
 
(3)   p + c = d + 6
 
 
solve the three equations :   (2) - (1) *2 :  4p + 2c + 2d - (2p+2c+2d)  =  80- 30*2
                                                                                                2p    =  20
                                                                                                  p    =  10
 
                                   p=10        plug in (3) :   c=d-4
                                   c=d-4       plug in (1) :   10 + (d -4) +d =30
                                                                                    6 +2d=30
                                                                                          d=12
                                   d=12        plug in (3)     10+c=12+6
                                                                                c=8
                                                                                

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