Answer to Question #91453 in Algebra for Ra

Question

3 major projects P₁, P₂, P₃, are being funded by 3 voluntary agencies V₁, V₂, V₃. V₁, V₂, V₃, are willing to pay Rs. 8,000/-, Rs. 4,000/- and Rs. 2, 000/-, respectively per person on the project P₁; Rs. 4,000/-, Rs. 3,000/- and Rs. 4, 000/- respectively per person on P₂; and Rs. 3,000/-, Rs. 5,000/-, Rs. 8,000/- respectively on the project P₃. Further, the amount that V₁, V₂, V₃, have kept aside for paying people on these projects is Rs. 2,17,000/-, Rs. 1,42,000/- and Rs. 1,32,000/- respectively. How many people should each project employ so that the total money available is utilised?

Answer 1

These equations reduces to

1. 8x+4y+3z=217

2. 4x+3y+5z=142

3. 2x+4y+8z=132

Now by multiplying 2nd equation by 2 and then subtract 1 from 2

That is

8x+6y+10z=284. (2nd equation×2)

-8x+4y+3z=217. (1st equation)

That is 2y+7z=67

y=(67-7z)/2. (p₁)

And multiply 2nd equation by 4 and 1st equation by 3 and subtract second from first equation

We get,

24x+12y+9z=651

-16x+12y+20z=568

We get

8x-11z=83

x=(83+11z)/8. ...…..(p₂)

Substituting values of x and y from p₂ and p₁ respectively in equation 3rd

That is

2x+4y+8z=132

2(83+11z)/8+4(67-7z)/2+8z=132

It reduces to

(83+11z)/4+134-14z+8z=132

(83+11z)/4+2=6z

83+11z+8=24z

Or 13z=91

z=7;

Substituting values of z into p₂ and p₁ ,we get

y=(67-7z)/2

y=(67-7×7)/2

y=18/2

y=9;

X=(83+11z)/8

X=(83+11×7)/8

X=160/8

X=20;

Hence we get x=20

y=9

z=7.