Given that x and y are positive integers and (x·2)+x(2^2)+(y·2^3)+(y·2^4)=42, what is the value of xy?
x=3, y=1
So the answer is 3
:)
(x·2)+x(2^2)+(y·2^3)+(y·2^4)=42
2x + 4x + 8y + 16y = 42
6x + 24y = 42
x + 4y = 7
x = 7 - 4y
so you are told that both x and y are POSITIVE integers,
what are the only possible values of y for x to be a positive integers
Let's simplify the given equation step by step:
(x·2) + x(2^2) + (y·2^3) + (y·2^4) = 42
Rewriting the powers of 2:
(x·2) + x(4) + (y·8) + (y·16) = 42
Now, combine like terms:
2x + 4x + 8y + 16y = 42
Simplifying further:
6x + 24y = 42
Dividing both sides by 6:
x + 4y = 7
Now, we need to express x in terms of y:
x = 7 - 4y
Substitute this value of x in the equation xy:
(7 - 4y)y = xy
Expanding and simplifying:
7y - 4y^2 = xy
Since we're given that x and y are positive integers, we know that x and y are both greater than zero. Therefore, xy must also be greater than zero.
To find the value of xy, we need to solve the equation 7y - 4y^2 = xy and find the positive integer solution.
Unfortunately, without any additional information or constraints, we cannot determine the specific value of xy.
To find the value of xy, we need to solve the given equation. Let's break it down step by step.
(x·2) + x(2^2) + (y·2^3) + (y·2^4) = 42
First, simplify the terms involving powers of 2:
2x + 4x + 8y + 16y = 42
Combining like terms:
6x + 24y = 42
Now, we want to find the possible values of x and y that satisfy this equation. Since x and y are positive integers, we can start by substituting different values and checking if they satisfy the equation.
Let's solve the equation for y in terms of x:
24y = 42 - 6x
y = (42 - 6x) / 24
y = (7 - x/4)
Since y is an integer, we need the right-hand side of the equation to be divisible by 4. Looking at the expression (7 - x/4), we can try substituting different values of x and see if it satisfies this condition.
Let's start with x = 4:
y = (7 - 4/4) = (7 - 1) = 6
Substituting x = 4 and y = 6 into the original equation:
(4·2) + 4(2^2) + (6·2^3) + (6·2^4) = 42
(8) + 4(4) + (6)(8) + (6)(16) = 42
8 + 16 + 48 + 96 = 42
The left-hand side of the equation is clearly larger than the right-hand side, so x = 4 and y = 6 are not the correct values.
Let's try another value of x, let's say x = 2:
y = (7 - 2/4) = (7 - 0.5) = 6.5
Since y is not an integer, this value of x is not valid.
Continuing this process, we can try different values of x until we find a valid solution that satisfies the equation. However, after trying different values, we find that there are no valid solutions for x and y that satisfy the equation.
Therefore, the value of xy cannot be determined.