Find the number of positive integer solutions for 4/x + 10/y=1
Answers
Step-by-step explanation:
Equation type: Ax + By = C
Few rules to find integral solutions of this type of equations.
First, reduce the equation in lowest reducible form.
After reducing, if coefficients of x and y still have a common factor, the equation will have no solutions.
If x and y are co-prime in the lowest reducible form, find any one integral solution. The rest of the solutions can be derived from that integral solution.
For each successive integral solutions of the equation, the value x and y will change by a coefficient of the other variable .If the equation is of the type Ax – By=C (after getting the lowest reducible form) ,an increase in x will cause increase in y. If the equation is of the type Ax + By=C,an increase in x will cause a decrease in y.
Let us take an example.
2x + 3y = 39.
(Number of Integral Solutions) Step-1: The equation is already in its reduced form and we can see that coefficients of x and y are co-prime.
(Number of Integral Solutions) Step-2: For a given equation, you should start substituting values (by hit and trial) for the variable that has larger coefficient to find out first integral solution. In this case, it is y. Now, if we take y = 0, we will get x = 39/2(not an integer). Again, if we take y=1, we will get x = 18. So, (18,1) is our first solution.
(Number of Integral Solutions) Step-3:If you understand the 4th point mentioned above, at one of any two consecutive integral values of y, the value of x will come out to be an integer OR at one of the 3 consecutive values of x, the value of y will come out to be an integer. That means, if we add 2n (where n is an integer) to the first value for y, we will have to subtract 3n from the first value of x to get integral solutions. That means,
If y =1 +2(1) = 3 , x= 18-3(1) = 15.
If y= 1 + 2(2) = 5, x= 18 – 3(2) = 12.
If y= 1 + 2(3) = 7, x = 18 – 3(3) = 9 and so on.
(Number of Integral Solutions) Step-4:This equation will have infinite number of integral solutions but finite number of non-negative integral solutions. Let’s see how we can find it.
We can keep increasing the value of y in the positive direction but x will be decreasing simultaneously and become less than 0 at one point. As lowest non negative integral value of y is 1,highest allowable positive value of x is 18 and it is decreasing by 3. So, x can take 7 non negative integral values and they are- 18, 15, 12, 9, 6, 3 and 0.Hence the given equation has 7 non negative integral values.