(x-y)(xr-xs-ys+yr)=[x+y][p-q]

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Solution for (x-y)(xr-xs-ys+yr)=[x+y][p-q] equation: 


Simplifying
(x + -1y)(xr + -1xs + -1ys + yr) = [x + y][p + -1q]

Reorder the terms:
(x + -1y)(rx + ry + -1sx + -1sy) = [x + y][p + -1q]

Multiply (x + -1y) * (rx + ry + -1sx + -1sy)
(x(rx + ry + -1sx + -1sy) + -1y * (rx + ry + -1sx + -1sy)) = [x + y][p + -1q]
((rx * x + ry * x + -1sx * x + -1sy * x) + -1y * (rx + ry + -1sx + -1sy)) = [x + y][p + -1q]

Reorder the terms:
((rxy + rx2 + -1sxy + -1sx2) + -1y * (rx + ry + -1sx + -1sy)) = [x + y][p + -1q]
((rxy + rx2 + -1sxy + -1sx2) + -1y * (rx + ry + -1sx + -1sy)) = [x + y][p + -1q]
(rxy + rx2 + -1sxy + -1sx2 + (rx * -1y + ry * -1y + -1sx * -1y + -1sy * -1y)) = [x + y][p + -1q]
(rxy + rx2 + -1sxy + -1sx2 + (-1rxy + -1ry2 + 1sxy + 1sy2)) = [x + y][p + -1q]

Reorder the terms:
(rxy + -1rxy + rx2 + -1ry2 + -1sxy + 1sxy + -1sx2 + 1sy2) = [x + y][p + -1q]

Combine like terms: rxy + -1rxy = 0
(0 + rx2 + -1ry2 + -1sxy + 1sxy + -1sx2 + 1sy2) = [x + y][p + -1q]
(rx2 + -1ry2 + -1sxy + 1sxy + -1sx2 + 1sy2) = [x + y][p + -1q]

Combine like terms: -1sxy + 1sxy = 0
(rx2 + -1ry2 + 0 + -1sx2 + 1sy2) = [x + y][p + -1q]
(rx2 + -1ry2 + -1sx2 + 1sy2) = [x + y][p + -1q]

Multiply [x + y] * [p + -1q]
rx2 + -1ry2 + -1sx2 + 1sy2 = [x[p + -1q] + y[p + -1q]]
rx2 + -1ry2 + -1sx2 + 1sy2 = [[p * x + -1q * x] + y[p + -1q]]
rx2 + -1ry2 + -1sx2 + 1sy2 = [[px + -1qx] + y[p + -1q]]
rx2 + -1ry2 + -1sx2 + 1sy2 = [px + -1qx + [p * y + -1q * y]]
rx2 + -1ry2 + -1sx2 + 1sy2 = [px + -1qx + [py + -1qy]]

Reorder the terms:
rx2 + -1ry2 + -1sx2 + 1sy2 = [px + py + -1qx + -1qy]
rx2 + -1ry2 + -1sx2 + 1sy2 = [px + py + -1qx + -1qy]

Solving
rx2 + -1ry2 + -1sx2 + 1sy2 = px + py + -1qx + -1qy

Solving for variable 'r'.

Move all terms containing r to the left, all other terms to the right.

Add 'sx2' to each side of the equation.
rx2 + -1ry2 + -1sx2 + sx2 + 1sy2 = px + py + -1qx + -1qy + sx2

Combine like terms: -1sx2 + sx2 = 0
rx2 + -1ry2 + 0 + 1sy2 = px + py + -1qx + -1qy + sx2
rx2 + -1ry2 + 1sy2 = px + py + -1qx + -1qy + sx2

Add '-1sy2' to each side of the equation.
rx2 + -1ry2 + 1sy2 + -1sy2 = px + py + -1qx + -1qy + sx2 + -1sy2

Combine like terms: 1sy2 + -1sy2 = 0
rx2 + -1ry2 + 0 = px + py + -1qx + -1qy + sx2 + -1sy2
rx2 + -1ry2 = px + py + -1qx + -1qy + sx2 + -1sy2

Reorder the terms:
-1px + -1py + qx + qy + rx2 + -1ry2 + -1sx2 + sy2 = px + py + -1qx + -1qy + sx2 + -1sy2 + -1px + -1py + qx + qy + -1sx2 + sy2

Reorder the terms:
-1px + -1py + qx + qy + rx2 + -1ry2 + -1sx2 + sy2 = px + -1px + py + -1py + -1qx + qx + -1qy + qy + sx2 + -1sx2 + -1sy2 + sy2

Combine like terms: px + -1px = 0
-1px + -1py + qx + qy + rx2 + -1ry2 + -1sx2 + sy2 = 0 + py + -1py + -1qx + qx + -1qy + qy + sx2 + -1sx2 + -1sy2 + sy2
-1px + -1py + qx + qy + rx2 + -1ry2 + -1sx2 + sy2 = py + -1py + -1qx + qx + -1qy + qy + sx2 + -1sx2 + -1sy2 + sy2

Combine like terms: py + -1py = 0
-1px + -1py + qx + qy + rx2 + -1ry2 + -1sx2 + sy2 = 0 + -1qx + qx + -1qy + qy + sx2 + -1sx2 + -1sy2 + sy2
-1px + -1py + qx + qy + rx2 + -1ry2 + -1sx2 + sy2 = -1qx + qx + -1qy + qy + sx2 + -1sx2 + -1sy2 + sy2

Combine like terms: -1qx + qx = 0
-1px + -1py + qx + qy + rx2 + -1ry2 + -1sx2 + sy2 = 0 + -1qy + qy + sx2 + -1sx2 + -1sy2 + sy2
-1px + -1py + qx + qy + rx2 + -1ry2 + -1sx2 + sy2 = -1qy + qy + sx2 + -1sx2 + -1sy2 + sy2

Combine like terms: -1qy + qy = 0
-1px + -1py + qx + qy + rx2 + -1ry2 + -1sx2 + sy2 = 0 + sx2 + -1sx2 + -1sy2 + sy2
-1px + -1py + qx + qy + rx2 + -1ry2 + -1sx2 + sy2 = sx2 + -1sx2 + -1sy2 + sy2

Combine like terms: sx2 + -1sx2 = 0
-1px + -1py + qx + qy + rx2 + -1ry2 + -1sx2 + sy2 = 0 + -1sy2 + sy2
-1px + -1py + qx + qy + rx2 + -1ry2 + -1sx2 + sy2 = -1sy2 + sy2

Combine like terms: -1sy2 + sy2 = 0
-1px + -1py + qx + qy + rx2 + -1ry2 + -1sx2 + sy2 = 0

The solution to this equation could not be determined.

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