if x,y are integral solution of 2x^2-3xy-2y^2=7 then value of |x+y| is:
To find the value of |x+y|, we can solve the given equation and then substitute the integral solutions of (x, y) into the expression |x+y|. Let's solve the equation step by step:
Step 1: Write the equation again:
2x^2 - 3xy - 2y^2 = 7
Step 2: Factorize the quadratic equation:
(2x + y)(x - 2y) = 7
Step 3: Find the factors of 7:
The factors of 7 are 1 and 7 (-1 and -7 for negative values).
Step 4: Set each factor equal to the expressions:
(2x + y) = 1, (x - 2y) = 7
Step 5: Solve the first equation:
2x + y = 1
y = 1 - 2x
Step 6: Substitute the value of y from the first equation into the second equation:
x - 2(1 - 2x) = 7
x - 2 + 4x = 7
5x = 7 + 2
5x = 9
x = 9/5
Step 7: Substitute the value of x into the first equation to find y:
y = 1 - 2(9/5)
y = 1 - 18/5
y = 5/5 - 18/5
y = -13/5
So, the integral solution is (x, y) = (9/5, -13/5).
Step 8: Calculate |x+y|:
|x+y| = |(9/5) + (-13/5)|
= |(9 - 13)/5|
= |-4/5|
= 4/5
Therefore, the value of |x+y| is 4/5.