Simplify ((a-y)/(a-2y))-((a+y)/(a+2y))

Math
(a-ya-2y)-(a+ya+2y)
To write a-ya-2y as a fraction with a common denominator, multiply by a+2ya+2y.
a-ya-2y⋅a+2ya+2y-a+ya+2y
To write -a+ya+2y as a fraction with a common denominator, multiply by a-2ya-2y.
a-ya-2y⋅a+2ya+2y-a+ya+2y⋅a-2ya-2y
Write each expression with a common denominator of (a-2y)(a+2y), by multiplying each by an appropriate factor of 1.
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Multiply a-ya-2y and a+2ya+2y.
(a-y)(a+2y)(a-2y)(a+2y)-a+ya+2y⋅a-2ya-2y
Multiply a+ya+2y and a-2ya-2y.
(a-y)(a+2y)(a-2y)(a+2y)-(a+y)(a-2y)(a+2y)(a-2y)
Reorder the factors of (a-2y)(a+2y).
(a-y)(a+2y)(a+2y)(a-2y)-(a+y)(a-2y)(a+2y)(a-2y)
(a-y)(a+2y)(a+2y)(a-2y)-(a+y)(a-2y)(a+2y)(a-2y)
Combine the numerators over the common denominator.
(a-y)(a+2y)-(a+y)(a-2y)(a+2y)(a-2y)
Simplify the numerator.
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Expand (a-y)(a+2y) using the FOIL Method.
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Apply the distributive property.
a(a+2y)-y(a+2y)-(a+y)(a-2y)(a+2y)(a-2y)
Apply the distributive property.
a⋅a+a(2y)-y(a+2y)-(a+y)(a-2y)(a+2y)(a-2y)
Apply the distributive property.
a⋅a+a(2y)-ya-y(2y)-(a+y)(a-2y)(a+2y)(a-2y)
a⋅a+a(2y)-ya-y(2y)-(a+y)(a-2y)(a+2y)(a-2y)
Simplify and combine like terms.
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Simplify each term.
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Multiply a by a.
a2+a(2y)-ya-y(2y)-(a+y)(a-2y)(a+2y)(a-2y)
Rewrite using the commutative property of multiplication.
a2+2ay-ya-y(2y)-(a+y)(a-2y)(a+2y)(a-2y)
Multiply y by y.
a2+2ay-ya-1⋅2y2-(a+y)(a-2y)(a+2y)(a-2y)
Multiply -1 by 2.
a2+2ay-ya-2y2-(a+y)(a-2y)(a+2y)(a-2y)
a2+2ay-ya-2y2-(a+y)(a-2y)(a+2y)(a-2y)
Subtract ya from 2ay.
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Move y.
a2+2ay-1ay-2y2-(a+y)(a-2y)(a+2y)(a-2y)
Subtract ay from 2ay.
a2+1ay-2y2-(a+y)(a-2y)(a+2y)(a-2y)
a2+1ay-2y2-(a+y)(a-2y)(a+2y)(a-2y)
a2+1ay-2y2-(a+y)(a-2y)(a+2y)(a-2y)
Multiply a by 1.
a2+ay-2y2-(a+y)(a-2y)(a+2y)(a-2y)
Apply the distributive property.
a2+ay-2y2+(-a-y)(a-2y)(a+2y)(a-2y)
Expand (-a-y)(a-2y) using the FOIL Method.
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Apply the distributive property.
a2+ay-2y2-a(a-2y)-y(a-2y)(a+2y)(a-2y)
Apply the distributive property.
a2+ay-2y2-a⋅a-a(-2y)-y(a-2y)(a+2y)(a-2y)
Apply the distributive property.
a2+ay-2y2-a⋅a-a(-2y)-ya-y(-2y)(a+2y)(a-2y)
a2+ay-2y2-a⋅a-a(-2y)-ya-y(-2y)(a+2y)(a-2y)
Simplify and combine like terms.
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Simplify each term.
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Multiply a by a by adding the exponents.
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Move a.
a2+ay-2y2-(a⋅a)-a(-2y)-ya-y(-2y)(a+2y)(a-2y)
Multiply a by a.
a2+ay-2y2-a2-a(-2y)-ya-y(-2y)(a+2y)(a-2y)
a2+ay-2y2-a2-a(-2y)-ya-y(-2y)(a+2y)(a-2y)
Rewrite using the commutative property of multiplication.
a2+ay-2y2-a2-1⋅-2(ay)-ya-y(-2y)(a+2y)(a-2y)
Multiply -1 by -2.
a2+ay-2y2-a2+2(ay)-ya-y(-2y)(a+2y)(a-2y)
Multiply y by y.
a2+ay-2y2-a2+2ay-ya-1⋅-2y2(a+2y)(a-2y)
Multiply -1 by -2.
a2+ay-2y2-a2+2ay-ya+2y2(a+2y)(a-2y)
a2+ay-2y2-a2+2ay-ya+2y2(a+2y)(a-2y)
Subtract ya from 2ay.
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Move y.
a2+ay-2y2-a2+2ay-1ay+2y2(a+2y)(a-2y)
Subtract ay from 2ay.
a2+ay-2y2-a2+1ay+2y2(a+2y)(a-2y)
a2+ay-2y2-a2+1ay+2y2(a+2y)(a-2y)
a2+ay-2y2-a2+1ay+2y2(a+2y)(a-2y)
Multiply a by 1.
a2+ay-2y2-a2+ay+2y2(a+2y)(a-2y)
Subtract a2 from a2.
ay-2y2+0+ay+2y2(a+2y)(a-2y)
Add ay and 0.
-2y2+ay+ay+2y2(a+2y)(a-2y)
Add -2y2 and 2y2.
ay+ay+0(a+2y)(a-2y)
Add ay+ay and 0.
ay+ay(a+2y)(a-2y)
Add ay and ay.
2ay(a+2y)(a-2y)
2ay(a+2y)(a-2y)
Simplify ((a-y)/(a-2y))-((a+y)/(a+2y))

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