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x+xy+y=64quad x+y=?

$x+x y+y=64 \quad x+y=?$

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Solve equations using square roots

Full solution

Q.

x+x y+y=64 \quad x+y=?

Factor out x: Factor out x from the terms involving x. $\newline$ $x(1+y) + y = 64$
Common factor substitution: Notice that $x+y$ is a common factor. $\newline$ Let's set $x+y$ equal to some variable, say $z$ . $\newline$ So, $x+y=z$ and $x= z-y$
Substitute $x$ with $z-y$ : Substitute $x$ with $z-y$ in the equation. $(z-y)(1+y) + y = 64$
Distribute the terms: Distribute the terms. $z + zy - y - y^2 + y = 64$
Combine like terms: Combine like terms. $z + zy - y^2 = 64$
Replace $zy$ with $xy$ : Since $z = x+y$ , replace $zy$ with $xy$ . $x+y + xy - y^2 = 64$
Simplify the equation: We know that $x+xy+y = 64$ , so by substitution: $\newline$ $x+y + (x+xy+y) - y^2 = 64$
Substitution and simplification: Simplify the equation. $x+y + 64 - y^2 = 64$
Final simplification: Subtract $64$ from both sides. $\newline$ $x+y - y^2 = 0$

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