What is the price per pound of apples, berries and cherries?

Question

 Mary pays $34 for 2 pounds of apples, 1 pound of berries, and 4 pounds of cherries. Tom pays $35 for 3 pounds of apples, 2 pounds of berries, and 2 pounds of cherries. Lee plays $49 for 5 pounds of apples, 3 pounds of berries, and 2 pounds of cherries. What is the price per pound for apples, berries and cherries?

Answer 1

34 = 2A + 1B + 4C

35 = 3A + 2B+ 2C

49 = 5A + 3B+ 2C

 

Pairs the 1st and 2nd equation, then eliminates C:

 34 = 2A + 1B + 4C

 35 = 3A + 2B + 2C

 

34   =  2A  +  1B + 4C

-70 = -6A + -4B + -4C  <---- multiplies 2nd equation by -2

 

Adds them together

-36 = -4A + -3B

36 = 4A + 3B  <----- please label this equation ALPHA

 

 

Next, pairs the 2nd and 3rd equations, and eliminates C:

35 = 3A + 2B+ 2C
49 = 5A + 3B+ 2C

 

Subtracts the top equation from the bottom:

49-35 = 5a-3a + 3b-2b + 2c-2c

14 = 2a + b <--- please label this equation BETA

 

 

ALPHA: 36 = 4A + 3B 

BETA:   14 = 2a + b

 

Solves the 2x2 system by substitution:

BETA: 14 - 2a = b <---- solves equation BETA for b

 

Substitutes into equation ALPHA:

36 = 4a + 3(14 - 2a)

36 = 4a + 42 - 6a

36 = 42 - 2a

-6 = -2a

a=3

 

b = 14 - 2a = 14 - 2(3) = 14 - 6 = 8

 

34 = 2a + b+ 4c

34 = 2(3)+ 8 + 4c

34 = 6+8 + 4c

34 = 14 + 4c

20 = 4c

c=5

 

1 pound of apples cost $3

1 pound of berries cost $8

1 pound of cherries cost $5