Question

Solve:
1) ax+by=c
bx+ay=1+c
2) A man rowing at the rate of 5km/h in still water takes thrice as much time in going 40km up the river as in going 40km down. Find the rate at which the river flows.
3) Some money is to be distributed equally among some children of a locality. if there are 8 children less, every one will get 10 rupees more and if there are 16 children more, every one will get 10 rupees less. What is the total amount of the money to be distributed?



I need STEP BY STEP answer so that I can understand it better. Please DONOT write the answer in a HURRY.
Would be very grateful if I get these answers by today. >_<

Answer 1

Hey there! ☺☻☺

1) Given,
ax + by = c ......(i)
bx + ay = 1+c......(ii)

Multiply eq. (i) by a:
a²x + aby = ac ........(iii)

Multiply eq. (ii) by b:
b²x + aby = b + bc  .......(iv)


Now,
Solving eq. (iii) & (iv):

a²x + aby = ac
b²x + aby = b + b
===============
(a² - b²)x = (a-b)c-b
x = {(a-b)c-b} / {a² - b²}
x = c/{a+b} - b/{a² - b²}

Now using eq. (i):
a [c/{a+b} - b/{a² - b²}] + by = c
by = c - ca/{a+b} + ab/{a² - b²}
by = {ca + cb-ca}/{a+b} + ab/{a² - b²}
y = c/a+b + a/a² - b²

We know,
x/a - y/b = 0
∴ bx - ay = 0.....(1)
ax+by = a² + b².......(2)
Now, Follow same as we did earlier:

Multiply eq. (1) by b:
b²x - aby = 0........(3)

Multiply eq. (2) by a:
a²x + aby = a(a²+b²).........(4)

Now, Solving eq. (3) and (4):
b²x - aby = 0
a²x + aby = a(a²+b²)
================
(a²+b²)x = a (a²+b²)
x = a

Now using (1):
ab - ay = 0
ay = ab
y = b

Hence, x= a and y = b.

2) Let the rate of river flow be x.
Time taken by man upstream = 40/5-x
Time taken by man downstream = 40/5+x
ATQ,
40/(5-x)= 3(40/5+x)
40(5+x)= 120 (5-x)
200+40x= 600 -120x
160x = 400
x = 2.5

Hence, rate of river flow = 2.5 km/h.

3) Let total amount be a.
And no. of children be x.

Now,
ATQ,
If there are 8 students less , everyone will get Rs.10 more.
=> a/(x-8)- a/x=10
Now, multiplying by x(x-8):
ax - a(x-8) = 10x(x-8) ax- ax+ 8a = 10x² - 80x8a = 10x² - 80x...........(i)

Now,
ATQ,
if there are 16 students more, each will get Rs.10 less:
a/x - a(x+16) = 10
Multiplying it by x(x+16):
a(x+16)- ax = 10x(x+16)
ax - 16a - ax = 10x² + 160x
16a = 10x² + 160x
8a = 5x² + 80x   [dividing by 2] ..........(ii)

Comparing eq. (i) and (ii):
Taking 8a as common:
10x² - 80x = 5x² + 80x
10x² - 5x² - 80x - 80x = 0
5x² - 160x = 0
5x - 160 = 0
5x = 160
x = 160/5
x = 32

Now, no. of children is 32.

Putting children value in eq.(ii):
8a = 5x² + 80x
8a = 5(32)² + 80×32
8a = 5120 + 2560
8a = 7680
a = 7680/8
a = 960

Hence, Total amount=a = Rs. 960.

Hope It Helps You! ☺☻☺