alternative to factoring

[rkav]

In question 64 on page 161 in OG 13th edition-

A fruit stand sold apples for .70 each and bananas for .50 each. If a customer purchased both apples and bananas from the stand for a total of 6.30 what total number of bananas and apples did the customer purchase?

The answer guide solves this problem by factoring to solve for y. I initially tried to tackle it by first solving for x but I seemed to have gone wrong somewhere and cant quite figure it out, Is this another way I can go about it or is factoring the only way?

.7x+.5y=6.30
7x+5y=63
7x=63-5y
x=9-(5y/7)

*plugging in x into original equation*

7(9-5y/7)+5y= 63
63-(35y/7)+5y=63
63-5y+5y=63


Everything cancelled out. Where did I go wrong? Thanks!

[Anju@Gurome]

*plugging in x into original equation*
...
Everything cancelled out. Where did I go wrong? Thanks!

Everything cancels out means the plugged value satisfy the equation.
It is obvious that if you get the value of x from an equation and then plugged the same value for x in the same equation, the value will always satisfy the equation, i.e. everything will cancel out.

A fruit stand sold apples for .70 each and bananas for .50 each. If a customer purchased both apples and bananas from the stand for a total of 6.30 what total number of bananas and apples did the customer purchase?

Let us assume the customer purchased x apples and y bananas, where x and y are positive integers.
Hence, (0.7)*x + (0.5)*y = 6.30

So, 7x + 5y = 63
--> x = (63 - 5y)/7

As x must be a positive integer, (63 - 5y) must be a multiple of 7.
As 63 is a multiple of 7, 5y must be a multiple of 7 ---> y must be a multiple of 7.

Take y = 7 ---> x = (63 - 5*7)/7 = 4
Take y = 14 ---> x = (63 - 5*14) < 0

Only possible value of y is 7.
Hence, x = 4 and y = 7.

[rkav]

Thank you! Now that makes sense!

[vipulgoyal]

alt approach

.7x+.5y =6.3
y could be 1,2,3....
if y is even
then .7x should divide 5.3,4.3,3.3,2.3,1.3
if y odd 1,3,.....
then .7x should divide 5.8,4.8,3.8,2.8,1.8,.8
bingo from here we get
6.3-2.8=3.5
6.3-.7*4= 7*.5