( m - n ) d y + ( 1 - y ) d x = 0
Solve for d (complex solution)
\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{C}\text{, }&\left(y=0\text{ and }x=0\right)\text{ or }\left(m=n+x-\frac{x}{y}\text{ and }y\neq 0\right)\end{matrix}\right.
Solve for m (complex solution)
\left\{\begin{matrix}m=n+x-\frac{x}{y}\text{, }&y\neq 0\\m\in \mathrm{C}\text{, }&d=0\text{ or }\left(x=0\text{ and }y=0\right)\end{matrix}\right.
Solve for d
\left\{\begin{matrix}\\d=0\text{, }&\text{unconditionally}\\d\in \mathrm{R}\text{, }&\left(y=0\text{ and }x=0\right)\text{ or }\left(m=n+x-\frac{x}{y}\text{ and }y\neq 0\right)\end{matrix}\right.
Solve for m
\left\{\begin{matrix}m=n+x-\frac{x}{y}\text{, }&y\neq 0\\m\in \mathrm{R}\text{, }&d=0\text{ or }\left(x=0\text{ and }y=0\right)\end{matrix}\right.
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\left(md-nd\right)y+\left(1-y\right)dx=0
Use the distributive property to multiply m-n by d.
mdy-ndy+\left(1-y\right)dx=0
Use the distributive property to multiply md-nd by y.
mdy-ndy+\left(d-yd\right)x=0
Use the distributive property to multiply 1-y by d.
mdy-ndy+dx-ydx=0
Use the distributive property to multiply d-yd by x.
\left(my-ny+x-yx\right)d=0
Combine all terms containing d.
\left(-xy+x+my-ny\right)d=0
The equation is in standard form.
d=0
Divide 0 by my-ny+x-yx.
\left(md-nd\right)y+\left(1-y\right)dx=0
Use the distributive property to multiply m-n by d.
mdy-ndy+\left(1-y\right)dx=0
Use the distributive property to multiply md-nd by y.
mdy-ndy+\left(d-yd\right)x=0
Use the distributive property to multiply 1-y by d.
mdy-ndy+dx-ydx=0
Use the distributive property to multiply d-yd by x.
mdy+dx-ydx=ndy
Add ndy to both sides. Anything plus zero gives itself.
mdy-ydx=ndy-dx
Subtract dx from both sides.
mdy=ndy-dx+ydx
Add ydx to both sides.
dym=dxy-dx+dny
The equation is in standard form.
\frac{dym}{dy}=\frac{d\left(xy-x+ny\right)}{dy}
Divide both sides by dy.
m=\frac{d\left(xy-x+ny\right)}{dy}
Dividing by dy undoes the multiplication by dy.
m=n+x-\frac{x}{y}
Divide d\left(ny-x+yx\right) by dy.
\left(md-nd\right)y+\left(1-y\right)dx=0
Use the distributive property to multiply m-n by d.
mdy-ndy+\left(1-y\right)dx=0
Use the distributive property to multiply md-nd by y.
mdy-ndy+\left(d-yd\right)x=0
Use the distributive property to multiply 1-y by d.
mdy-ndy+dx-ydx=0
Use the distributive property to multiply d-yd by x.
\left(my-ny+x-yx\right)d=0
Combine all terms containing d.
\left(-xy+x+my-ny\right)d=0
The equation is in standard form.
d=0
Divide 0 by my-ny+x-yx.
\left(md-nd\right)y+\left(1-y\right)dx=0
Use the distributive property to multiply m-n by d.
mdy-ndy+\left(1-y\right)dx=0
Use the distributive property to multiply md-nd by y.
mdy-ndy+\left(d-yd\right)x=0
Use the distributive property to multiply 1-y by d.
mdy-ndy+dx-ydx=0
Use the distributive property to multiply d-yd by x.
mdy+dx-ydx=ndy
Add ndy to both sides. Anything plus zero gives itself.
mdy-ydx=ndy-dx
Subtract dx from both sides.
mdy=ndy-dx+ydx
Add ydx to both sides.
dym=dxy-dx+dny
The equation is in standard form.
\frac{dym}{dy}=\frac{d\left(xy-x+ny\right)}{dy}
Divide both sides by dy.
m=\frac{d\left(xy-x+ny\right)}{dy}
Dividing by dy undoes the multiplication by dy.
m=n+x-\frac{x}{y}
Divide d\left(ny-x+yx\right) by dy.
Examples
Quadratic equation
{ x } ^ { 2 } - 4 x - 5 = 0
Trigonometry
4 \sin \theta \cos \theta = 2 \sin \theta
Linear equation
y = 3x + 4
Arithmetic
699 * 533
Matrix
\left[ \begin{array} { l l } { 2 } & { 3 } \\ { 5 } & { 4 } \end{array} \right] \left[ \begin{array} { l l l } { 2 } & { 0 } & { 3 } \\ { -1 } & { 1 } & { 5 } \end{array} \right]
Simultaneous equation
\left. \begin{cases} { 8x+2y = 46 } \\ { 7x+3y = 47 } \end{cases} \right.
Differentiation
\frac { d } { d x } \frac { ( 3 x ^ { 2 } - 2 ) } { ( x - 5 ) }
Integration
\int _ { 0 } ^ { 1 } x e ^ { - x ^ { 2 } } d x
Limits
\lim _{x \rightarrow-3} \frac{x^{2}-9}{x^{2}+2 x-3}