Solving x/a y/b=2 and ax-by=a square- b square
Step 1: Cross-multiply x/a y/b=2 to eliminate fractions
To eliminate the fractions in the first equation, we can cross-multiply both sides by the denominators:
x/a * a/b = 2 * a/b
After simplification:
xb = 2a
So, the first equation can be written as:
xb = 2a
Step 2: Simplify the second equation
Now, let's simplify the second equation:
ax - by = a^2 - b^2
We can rewrite this equation as:
ax - a^2 = by - b^2
Factor out a from the left-hand side and b from the right-hand side:
a(x - a) = b(b - y)
So, the second equation can be written as:
a(x - a) = b(b - y)
Step 3: Solve for y in terms of x
Now, let's solve for y in terms of x:
xb = 2a
y = (2a - bx) / b
Step 4: Substitute y into the second equation
Substituting y into the second equation:
a(x - a) = b(b - (2a - bx) / b)
Simplifying:
a(x - a) = b^2 - 2a + bx
Expanding:
ax - a^2 = b^2 - 2a + bx
Bringing all the terms to one side:
bx - ax + a^2 - b^2 + 2a = 0
Simplifying:
(b - a)x + (a + b)(a - b) = 0
Factor out (b - a):
(b - a)(x - (a + b)) = 0
Step 5: Solve for x and y
So, the solutions are:
x = a + b or y = (2a - bx) / b
If x = a + b:
y = (2a - b(a + b)) / b = (a - b) / b
If y = (2a - bx) / b:
x = (2a - by) / b = (2a - b(2a - bx)) / b = (3a - bx) / b