Sangeetha Pulapaka
1

We have a system of three linear equations in three variables

Let x represent each apple, y represent each blueberry , and z represent each cherry.

You have picked up 24 pounds of there different types of fruit at a farm. The linear equation which represents this is x+y+z =24 \rightarrow equation 1.

Twice the many pound of apples as the other two fruits combined can be represented by the linear equation

x = 2(y+z)\rightarrow equation 2.

The apples cost $1.40 per pound, blueberries cost$0.90 per pound, cherries cost $1.10 per pound which costs$30. The linear equation which represents this will be

1.40 x + 0.9 y + 1.1 z = 30\rightarrowequation 3.

Now we solve this by good old elimination and substitution method.

Solving equation 1 and 2,

x+ y + z = 24

x -2y-2z = 0

-----------------------------------

2x+2y+2z = 48 (multiplying equation 1 by 2)

x-2y-2z = 0

-------------------------------

3x = 48

\Rightarrow x = 16

Now plug in this value of x in equation 1 to get y+z = 24 - 16 = 8

and equation 3 to get 1.4 \times 16 + 0.9 y + 1.1 z = 30 or 0.9 y + 1.1 z = 30 - 22.4 = 7.6

Multiplying  y+z = 8 with 0.9 we get

0.9y + 0.9 z = 7.2

\mp 0.9 y \mp 1.1 z = \mp 7.6

----------------------------------------------

-0.2 z = -0.4

z = 2

Plugging the values of x and z in equation 1,  we get y = 24 - 16 - 2 = 24 - 18 = 6

So, the number of apples are 16, the number of blueberries are 6, and the number of cherries are 2.

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