Solve for x, y (complex solution)
\left\{\begin{matrix}x=\frac{c\left(a+b\right)}{b^{2}}\text{, }y=\frac{c}{b}\text{, }&b\neq 0\\x\in \mathrm{C}\text{, }y=0\text{, }&c=0\text{ and }b=0\\x\in \mathrm{C}\text{, }y\in \mathrm{C}\text{, }&a=0\text{ and }b=0\text{ and }c=0\end{matrix}\right.
Solve for x, y
\left\{\begin{matrix}x=\frac{c\left(a+b\right)}{b^{2}}\text{, }y=\frac{c}{b}\text{, }&b\neq 0\\x\in \mathrm{R}\text{, }y=0\text{, }&c=0\text{ and }b=0\\x\in \mathrm{R}\text{, }y\in \mathrm{R}\text{, }&a=0\text{ and }b=0\text{ and }c=0\end{matrix}\right.
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by=c,\left(-a\right)y+bx=c
To solve a pair of equations using substitution, first solve one of the equations for one of the variables. Then substitute the result for that variable in the other equation.
by=c
Pick one of the two equations which is more simple to solve for y by isolating y on the left hand side of the equal sign.
y=\frac{c}{b}
Divide both sides by b.
\left(-a\right)\times \frac{c}{b}+bx=c
Substitute \frac{c}{b} for y in the other equation, \left(-a\right)y+bx=c.
-\frac{ac}{b}+bx=c
Multiply -a times \frac{c}{b}.
bx=\frac{c\left(a+b\right)}{b}
Add \frac{ac}{b} to both sides of the equation.
x=\frac{c\left(a+b\right)}{b^{2}}
Divide both sides by b.
y=\frac{c}{b},x=\frac{c\left(a+b\right)}{b^{2}}
The system is now solved.
by=c,\left(-a\right)y+bx=c
To solve a pair of equations using substitution, first solve one of the equations for one of the variables. Then substitute the result for that variable in the other equation.
by=c
Pick one of the two equations which is more simple to solve for y by isolating y on the left hand side of the equal sign.
y=\frac{c}{b}
Divide both sides by b.
\left(-a\right)\times \frac{c}{b}+bx=c
Substitute \frac{c}{b} for y in the other equation, \left(-a\right)y+bx=c.
-\frac{ac}{b}+bx=c
Multiply -a times \frac{c}{b}.
bx=\frac{c\left(a+b\right)}{b}
Add \frac{ac}{b} to both sides of the equation.
x=\frac{c\left(a+b\right)}{b^{2}}
Divide both sides by b.
y=\frac{c}{b},x=\frac{c\left(a+b\right)}{b^{2}}
The system is now solved.
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}