The simplified expression is [tex]\frac{1 - x - y}{x + y}[/tex]and the restriction is [tex]y \ne -x[/tex]
How to simplify the expression?
The expression is given as:
[tex]\frac{x - y}{4x^2 - 8xy + 3y^2} \div \frac{2x + y}{2x - 3y} \times \frac{4x^2 - y^2}{x^2 - y^2} -1[/tex]
Express x^2 - y^2 as (x + y)(x - y) and factorize other expressions
[tex]\frac{x - y}{(2x - y)(2x - 3y)} \div \frac{2x + y}{2x - 3y} \times \frac{4x^2 - y^2}{(x - y)(x + y)} -1[/tex]
Rewrite the expression as products
[tex]\frac{x - y}{(2x - y)(2x - 3y)} \times \frac{2x - 3y}{2x + y} \times \frac{4x^2 - y^2}{(x - y)(x + y)} -1[/tex]
Cancel out the common factors
[tex]\frac{1}{(2x - y)} \times \frac{1}{2x + y} \times \frac{4x^2 - y^2}{(x + y)} -1[/tex]
Express 4x^2 - y^2 as (2x - y)(2x + y)
[tex]\frac{1}{(2x - y)} \times \frac{1}{2x + y} \times \frac{(2x - y)(2x + y)}{(x + y)} -1[/tex]
Cancel out the common factors
[tex]\frac{1}{x + y} -1[/tex]
Take the LCM
[tex]\frac{1 - x - y}{x + y}[/tex]
Hence, the simplified expression is [tex]\frac{1 - x - y}{x + y}[/tex]and the restriction is [tex]y \ne -x[/tex]
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