The number of non-negative solutions of the equation 12x+7y =35 is?
To find the number of non-negative solutions of the equation 12x + 7y = 35, we can use the technique of generating functions.
Firstly, let's write the equation in terms of generating functions. We express the equation as a power series: (1 + x^12 + x^24 + x^36 + ...) * (1 + x^7 + x^14 + x^21 + ...).
We can simplify this expression by dividing both sides of the equation by (1 - x^12) * (1 - x^7). This gives us:
(1 - x^12)^(-1) * (1 - x^7)^(-1)
Using the formula for the geometric series, we can rewrite the expression as:
(1 + x^12 + x^24 + x^36 + ...) * (1 + x^7 + x^14 + x^21 + ...)
Now, we need to find the coefficient of x^35 in this expression.
By multiplying the power series, we get terms of x^n when the exponents from the two power series sum up to n. So, we can find the coefficient of x^35 by looking at the terms with x^35 in it.
To obtain an x^35 term, we need to multiply (x^23 from the first series) by (x^12 from the second series) and add this term to the product of (x^11 from the first series) and (x^24 from the second series). This can be written as:
(x^23 * x^12) + (x^11 * x^24)
Simplifying this expression, we get:
x^35 + x^35 = 2x^35
Therefore, the coefficient of x^35 in the expression is 2.
Hence, there are 2 non-negative solutions to the equation 12x + 7y = 35.