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exttt{ $-2$ }( $7$ y - x) /y = $2$ x

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Factor sums and differences of cubes

Full solution

Q. exttt{

-2

}(

7

y - x) /y =

2

x

Recognize formula: First, recognize the sum of cubes formula: $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$ .
Apply formula to $t^3 + u^3$ : Apply the formula to $t^3 + u^3$ , setting $a = t$ and $b = u$ .
Factorization: The factorization becomes $(t + u)(t^2 - tu + u^2)$ .
Complete factorization: The missing binomial that completes the factorization is $t + u$ .
Multiply both sides: Multiply both sides by $y$ to get rid of the denominator: $-2(7y - x) = 2xy$ .
Distribute $-2$ : Distribute the $-2$ on the left side: $-14y + 2x = 2xy$ .
Subtract $2x$ : Subtract $2x$ from both sides to get all $x$ terms on one side: $-14y = 2xy - 2x$ .
Factor out common factor: Factor out the common factor of $2x$ on the right side: $-14y = 2x(y - 1)$ .
Divide both sides: Divide both sides by $2x$ to solve for $y$ : $y = -14y / (2x)$ .

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