HallsofIvy said:Where did the d in a/d come from? You can't just "stick" it into part of the equation.
Saying "d divides a" means a= dn for some integer n. Saying "d divides b" means b= dm for some integer m. Replace a and b in your equation by that and see what happens.
When there is no integral solution for x and y when c is not divisible by d, it means that there are no whole number values for x and y that satisfy the equation c = dx + dy. In other words, there is no combination of x and y that can be added together to equal c when d does not evenly divide into c.
Finding integral solutions for equations involving c and d is important because it helps us understand the relationships between numbers and how they can be combined to equal a specific value. It also allows us to solve real-world problems and make accurate calculations.
If c is divisible by d in an equation involving x and y, there will be an infinite number of integral solutions for x and y. This is because any multiple of d can be divided evenly by d, so there are an infinite number of combinations of x and y that can be added together to equal c.
No, there cannot be non-integer solutions for x and y when c is not divisible by d. This is because the definition of an integral solution is a solution in which both x and y are whole numbers. If c is not divisible by d, there are no whole number values that can satisfy the equation c = dx + dy.
We can determine if there are integral solutions for x and y when c is not divisible by d by using the Euclidean algorithm. This algorithm helps us find the greatest common divisor of two numbers, which can then help us determine if there are any integral solutions for x and y. If the greatest common divisor of c and d is not equal to 1, then there are no integral solutions. If the greatest common divisor is 1, then there will be integral solutions for x and y.