If x and y are nonnegative integers and 5x + 2y = 32, how many values

CONCEPT: Find one solution i.e. values of x and y and for each next solution remember that x always changes by co-efficient of y (i.e. 2 in this case) and y always changes by co-efficient of x (i.e. 5 in this case) and

First solution is x = 0 and y = 16
Second solution is x = 2 and y = 11
Third solution is x = 4 and y = 6
Third solution is x = 6 and y = 1

Answer: Option D

I am not able to understand any of the solutions for the given problem. Can someone explain it?

This is a liner equation question and you have to find possible values of x which satisfy the linear equation: 5x + 2y = 32

What this means is - how many non-negative values can x take, which satisfy this equation.
Also keep in mind y can take non-negative values only. i.e. integer values of 0, 1, 2, 3, ...

If we replace x in the equation by each of 0, 2, 4 and 6, we get corresponding values of y as 16, 11, 6 & 1.
In this manner:

when x=0, 5x = 0 and thus 2y=32, y=16 ---> 16 is not an integer and thus x can take value of 0.
when x=1, 5x = 5 and thus 2y=27, y=13.5 ---> 13.5 is not an integer and thus, x cannot take value of 1.
(taking only even values of x)
when x=2, 5x = 10 and thus 2y=22, y=11
when x=4, 5x = 20 and thus 2y=12, y=6
when x=6, 5x = 30 and thus 2y=2, y=2

Thus, x can take 4 values.
Ans: D

Hope this helps!

Solution:

Simplifying, we have:

5x = 32 - 2y

5x = 2(16 - y)

x = 2(16 - y)/5

The expression (16 - y) must be a multiple of 5, in order for the entire expression on the right side of the equation to be a non-negative integer. Thus, we see that y can be 1, 6, 11, or 16. Since there are 4 possible values for y, there are also 4 possible values for x.

Alternate Solution:

Notice x and y are nonnegative integers, we see that in order for 5x + 2y = 32 to hold, x can’t be greater than 6 (since 5 * 7 = 35 already). Therefore, x can only be any integers from 0 to 6, inclusive. Furthermore, if x is odd, then 5x is odd and 2y = 32 - 5x will be odd. However, since 2y is always even regardless of what y is, we see that x can’t be odd. On the other hand, if x is even, then 5x is even and 2y = 32 - 5x will be even. Therefore, x can only be 0, 2, 4 or 6,

Answer: D

The maximum value of x is 6 because 5x=30
Since 2y is always an even number 5x also has to be an even number
Therefore x can be 0,2,4,6
There are four values fore x

Answer D

5x + 2y = 32

Using Aryabhatta's method/ Chinese Remainder Theorem,

2y = 32 - 5x

Hence, \(y\) = \(16 - 5*\frac{x}{2}\)

now, since we are using non-negative integers, start from x = 0, till y >= 0, and substitue values only where x is divisible by 2

i.e.,
x = 0, y = 16
x = 2, y = 11
x = 4, y = 6
x = 6, y = 1

going any further will give us y < 0

Hence, 4 values, i.e., option D