(y-2)(y-1)=2(y-2)(y-3)

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Solution for (y-2)(y-1)=2(y-2)(y-3) equation: 


Simplifying
(y + -2)(y + -1) = 2(y + -2)(y + -3)

Reorder the terms:
(-2 + y)(y + -1) = 2(y + -2)(y + -3)

Reorder the terms:
(-2 + y)(-1 + y) = 2(y + -2)(y + -3)

Multiply (-2 + y) * (-1 + y)
(-2(-1 + y) + y(-1 + y)) = 2(y + -2)(y + -3)
((-1 * -2 + y * -2) + y(-1 + y)) = 2(y + -2)(y + -3)
((2 + -2y) + y(-1 + y)) = 2(y + -2)(y + -3)
(2 + -2y + (-1 * y + y * y)) = 2(y + -2)(y + -3)
(2 + -2y + (-1y + y2)) = 2(y + -2)(y + -3)

Combine like terms: -2y + -1y = -3y
(2 + -3y + y2) = 2(y + -2)(y + -3)

Reorder the terms:
2 + -3y + y2 = 2(-2 + y)(y + -3)

Reorder the terms:
2 + -3y + y2 = 2(-2 + y)(-3 + y)

Multiply (-2 + y) * (-3 + y)
2 + -3y + y2 = 2(-2(-3 + y) + y(-3 + y))
2 + -3y + y2 = 2((-3 * -2 + y * -2) + y(-3 + y))
2 + -3y + y2 = 2((6 + -2y) + y(-3 + y))
2 + -3y + y2 = 2(6 + -2y + (-3 * y + y * y))
2 + -3y + y2 = 2(6 + -2y + (-3y + y2))

Combine like terms: -2y + -3y = -5y
2 + -3y + y2 = 2(6 + -5y + y2)
2 + -3y + y2 = (6 * 2 + -5y * 2 + y2 * 2)
2 + -3y + y2 = (12 + -10y + 2y2)

Solving
2 + -3y + y2 = 12 + -10y + 2y2

Solving for variable 'y'.

Reorder the terms:
2 + -12 + -3y + 10y + y2 + -2y2 = 12 + -10y + 2y2 + -12 + 10y + -2y2

Combine like terms: 2 + -12 = -10
-10 + -3y + 10y + y2 + -2y2 = 12 + -10y + 2y2 + -12 + 10y + -2y2

Combine like terms: -3y + 10y = 7y
-10 + 7y + y2 + -2y2 = 12 + -10y + 2y2 + -12 + 10y + -2y2

Combine like terms: y2 + -2y2 = -1y2
-10 + 7y + -1y2 = 12 + -10y + 2y2 + -12 + 10y + -2y2

Reorder the terms:
-10 + 7y + -1y2 = 12 + -12 + -10y + 10y + 2y2 + -2y2

Combine like terms: 12 + -12 = 0
-10 + 7y + -1y2 = 0 + -10y + 10y + 2y2 + -2y2
-10 + 7y + -1y2 = -10y + 10y + 2y2 + -2y2

Combine like terms: -10y + 10y = 0
-10 + 7y + -1y2 = 0 + 2y2 + -2y2
-10 + 7y + -1y2 = 2y2 + -2y2

Combine like terms: 2y2 + -2y2 = 0
-10 + 7y + -1y2 = 0

Factor a trinomial.
(-5 + y)(2 + -1y) = 0


Subproblem 1
Set the factor '(-5 + y)' equal to zero and attempt to solve:

Simplifying
-5 + y = 0

Solving
-5 + y = 0

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

Add '5' to each side of the equation.
-5 + 5 + y = 0 + 5

Combine like terms: -5 + 5 = 0
0 + y = 0 + 5
y = 0 + 5

Combine like terms: 0 + 5 = 5
y = 5

Simplifying
y = 5

Subproblem 2
Set the factor '(2 + -1y)' equal to zero and attempt to solve:

Simplifying
2 + -1y = 0

Solving
2 + -1y = 0

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

Add '-2' to each side of the equation.
2 + -2 + -1y = 0 + -2

Combine like terms: 2 + -2 = 0
0 + -1y = 0 + -2
-1y = 0 + -2

Combine like terms: 0 + -2 = -2
-1y = -2

Divide each side by '-1'.
y = 2

Simplifying
y = 2
Solution
y = {5, 2}

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