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Simplifying (x + y)(2x + -1y) + -1(3x + -1y)(y + x) = 0 Multiply (x + y) * (2x + -1y) (x(2x + -1y) + y(2x + -1y)) + -1(3x + -1y)(y + x) = 0 ((2x * x + -1y * x) + y(2x + -1y)) + -1(3x + -1y)(y + x) = 0 Reorder the terms: ((-1xy + 2x2) + y(2x + -1y)) + -1(3x + -1y)(y + x) = 0 ((-1xy + 2x2) + y(2x + -1y)) + -1(3x + -1y)(y + x) = 0 (-1xy + 2x2 + (2x * y + -1y * y)) + -1(3x + -1y)(y + x) = 0 (-1xy + 2x2 + (2xy + -1y2)) + -1(3x + -1y)(y + x) = 0 Reorder the terms: (-1xy + 2xy + 2x2 + -1y2) + -1(3x + -1y)(y + x) = 0 Combine like terms: -1xy + 2xy = 1xy (1xy + 2x2 + -1y2) + -1(3x + -1y)(y + x) = 0 Reorder the terms: 1xy + 2x2 + -1y2 + -1(3x + -1y)(x + y) = 0 Multiply (3x + -1y) * (x + y) 1xy + 2x2 + -1y2 + -1(3x * (x + y) + -1y * (x + y)) = 0 1xy + 2x2 + -1y2 + -1((x * 3x + y * 3x) + -1y * (x + y)) = 0 Reorder the terms: 1xy + 2x2 + -1y2 + -1((3xy + 3x2) + -1y * (x + y)) = 0 1xy + 2x2 + -1y2 + -1((3xy + 3x2) + -1y * (x + y)) = 0 1xy + 2x2 + -1y2 + -1(3xy + 3x2 + (x * -1y + y * -1y)) = 0 1xy + 2x2 + -1y2 + -1(3xy + 3x2 + (-1xy + -1y2)) = 0 Reorder the terms: 1xy + 2x2 + -1y2 + -1(3xy + -1xy + 3x2 + -1y2) = 0 Combine like terms: 3xy + -1xy = 2xy 1xy + 2x2 + -1y2 + -1(2xy + 3x2 + -1y2) = 0 1xy + 2x2 + -1y2 + (2xy * -1 + 3x2 * -1 + -1y2 * -1) = 0 1xy + 2x2 + -1y2 + (-2xy + -3x2 + 1y2) = 0 Reorder the terms: 1xy + -2xy + 2x2 + -3x2 + -1y2 + 1y2 = 0 Combine like terms: 1xy + -2xy = -1xy -1xy + 2x2 + -3x2 + -1y2 + 1y2 = 0 Combine like terms: 2x2 + -3x2 = -1x2 -1xy + -1x2 + -1y2 + 1y2 = 0 Combine like terms: -1y2 + 1y2 = 0 -1xy + -1x2 + 0 = 0 -1xy + -1x2 = 0 Solving -1xy + -1x2 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '-1x'. -1x(y + x) = 0 Ignore the factor -1.Subproblem 1
Set the factor 'x' equal to zero and attempt to solve: Simplifying x = 0 Solving x = 0 Move all terms containing x to the left, all other terms to the right. Simplifying x = 0Subproblem 2
Set the factor '(y + x)' equal to zero and attempt to solve: Simplifying y + x = 0 Reorder the terms: x + y = 0 Solving x + y = 0 Move all terms containing x to the left, all other terms to the right. Add '-1y' to each side of the equation. x + y + -1y = 0 + -1y Combine like terms: y + -1y = 0 x + 0 = 0 + -1y x = 0 + -1y Remove the zero: x = -1y Simplifying x = -1ySolution
x = {0, -1y}
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