Mister Exam
Expression simplification/
Least common denominator/
(x*y+10*y^2/2-5*(y-1)^2-(x-10)*(y-1))/((x-10)*(y-1)+10*(y-1)^2/2-5*(y-2)^2-(x-20)*(y-2))
Least common denominator (x*y+10*y^2/2-5*(y-1)^2-(x-10)*(y-1))/((x-10)*(y-1)+10*(y-1)^2/2-5*(y-2)^2-(x-20)*(y-2))
The solution
You have entered
[src]
2
10*y 2
x*y + ----- - 5*(y - 1) - (x - 10)*(y - 1)
2
--------------------------------------------------------------
2
10*(y - 1) 2
(x - 10)*(y - 1) + ----------- - 5*(y - 2) - (x - 20)*(y - 2)
2
$$\frac{- \left(x - 10\right) \left(y - 1\right) + \left(- 5 \left(y - 1\right)^{2} + \left(x y + \frac{10 y^{2}}{2}\right)\right)}{- \left(x - 20\right) \left(y - 2\right) + \left(- 5 \left(y - 2\right)^{2} + \left(\left(x - 10\right) \left(y - 1\right) + \frac{10 \left(y - 1\right)^{2}}{2}\right)\right)}$$
(x*y + (10*y^2)/2 - 5*(y - 1)^2 - (x - 10)*(y - 1))/((x - 10)*(y - 1) + (10*(y - 1)^2)/2 - 5*(y - 2)^2 - (x - 20)*(y - 2))
General simplification
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-15 + x + 20*y -------------- -45 + x + 20*y
$$\frac{x + 20 y - 15}{x + 20 y - 45}$$
(-15 + x + 20*y)/(-45 + x + 20*y)
Rational denominator
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2 2
-10 + x - 5*(-1 + y) + 5*y + 10*y
-----------------------------------
-45 + x + 20*y
$$\frac{x + 5 y^{2} + 10 y - 5 \left(y - 1\right)^{2} - 10}{x + 20 y - 45}$$
(-10 + x - 5*(-1 + y)^2 + 5*y^2 + 10*y)/(-45 + x + 20*y)
Numerical answer
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(5.0*y^2 - 5.0*(-1.0 + y)^2 + x*y - (-1.0 + y)*(-10.0 + x))/(5.0*(-1.0 + y)^2 - 20.0*(-1 + 0.5*y)^2 + (-1.0 + y)*(-10.0 + x) - (-2.0 + y)*(-20.0 + x))
(5.0*y^2 - 5.0*(-1.0 + y)^2 + x*y - (-1.0 + y)*(-10.0 + x))/(5.0*(-1.0 + y)^2 - 20.0*(-1 + 0.5*y)^2 + (-1.0 + y)*(-10.0 + x) - (-2.0 + y)*(-20.0 + x))
Combinatorics
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-15 + x + 20*y -------------- -45 + x + 20*y
$$\frac{x + 20 y - 15}{x + 20 y - 45}$$
(-15 + x + 20*y)/(-45 + x + 20*y)
Assemble expression
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2 2
- 5*(-1 + y) + 5*y + x*y - (-1 + y)*(-10 + x)
---------------------------------------------------------------------
2 2
- 5*(-2 + y) + 5*(-1 + y) + (-1 + y)*(-10 + x) - (-20 + x)*(-2 + y)
$$\frac{x y + 5 y^{2} - \left(x - 10\right) \left(y - 1\right) - 5 \left(y - 1\right)^{2}}{- \left(x - 20\right) \left(y - 2\right) + \left(x - 10\right) \left(y - 1\right) - 5 \left(y - 2\right)^{2} + 5 \left(y - 1\right)^{2}}$$
(-5*(-1 + y)^2 + 5*y^2 + x*y - (-1 + y)*(-10 + x))/(-5*(-2 + y)^2 + 5*(-1 + y)^2 + (-1 + y)*(-10 + x) - (-20 + x)*(-2 + y))
Combining rational expressions
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2
- 5*(-1 + y) + y*(x + 5*y) - (-1 + y)*(-10 + x)
-------------------------------------------------------------
2
- 5*(-2 + y) + (-1 + y)*(-15 + x + 5*y) - (-20 + x)*(-2 + y)
$$\frac{y \left(x + 5 y\right) - \left(x - 10\right) \left(y - 1\right) - 5 \left(y - 1\right)^{2}}{- \left(x - 20\right) \left(y - 2\right) - 5 \left(y - 2\right)^{2} + \left(y - 1\right) \left(x + 5 y - 15\right)}$$
(-5*(-1 + y)^2 + y*(x + 5*y) - (-1 + y)*(-10 + x))/(-5*(-2 + y)^2 + (-1 + y)*(-15 + x + 5*y) - (-20 + x)*(-2 + y))
Common denominator
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30
1 + --------------
-45 + x + 20*y
$$1 + \frac{30}{x + 20 y - 45}$$
1 + 30/(-45 + x + 20*y)
Powers
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2 2
- 5*(-1 + y) + 5*y + x*y - (-1 + y)*(-10 + x)
---------------------------------------------------------------------
2 2
- 5*(-2 + y) + 5*(-1 + y) + (-1 + y)*(-10 + x) - (-20 + x)*(-2 + y)
$$\frac{x y + 5 y^{2} - \left(x - 10\right) \left(y - 1\right) - 5 \left(y - 1\right)^{2}}{- \left(x - 20\right) \left(y - 2\right) + \left(x - 10\right) \left(y - 1\right) - 5 \left(y - 2\right)^{2} + 5 \left(y - 1\right)^{2}}$$
(-5*(-1 + y)^2 + 5*y^2 + x*y - (-1 + y)*(-10 + x))/(-5*(-2 + y)^2 + 5*(-1 + y)^2 + (-1 + y)*(-10 + x) - (-20 + x)*(-2 + y))
Trigonometric part
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2 2
- 5*(-1 + y) + 5*y + x*y - (-1 + y)*(-10 + x)
---------------------------------------------------------------------
2 2
- 5*(-2 + y) + 5*(-1 + y) + (-1 + y)*(-10 + x) - (-20 + x)*(-2 + y)
$$\frac{x y + 5 y^{2} - \left(x - 10\right) \left(y - 1\right) - 5 \left(y - 1\right)^{2}}{- \left(x - 20\right) \left(y - 2\right) + \left(x - 10\right) \left(y - 1\right) - 5 \left(y - 2\right)^{2} + 5 \left(y - 1\right)^{2}}$$
(-5*(-1 + y)^2 + 5*y^2 + x*y - (-1 + y)*(-10 + x))/(-5*(-2 + y)^2 + 5*(-1 + y)^2 + (-1 + y)*(-10 + x) - (-20 + x)*(-2 + y))