\left\{ \begin{array} { l } { P _ { x } \cdot x + P _ { y } \cdot y = M } \\ { x = \frac { 1 } { 2 } \cdot \frac { H } { P _ { x } } } \end{array} \right.
Solve for x, y (complex solution)
\left\{\begin{matrix}x=\frac{H}{2P_{x}}\text{, }y=-\frac{H-2M}{2P_{y}}\text{, }&P_{y}\neq 0\text{ and }P_{x}\neq 0\\x=\frac{H}{2P_{x}}\text{, }y\in \mathrm{C}\text{, }&P_{x}\neq 0\text{ and }M=\frac{H}{2}\text{ and }P_{y}=0\end{matrix}\right.
Solve for x, y
\left\{\begin{matrix}x=\frac{H}{2P_{x}}\text{, }y=-\frac{H-2M}{2P_{y}}\text{, }&P_{y}\neq 0\text{ and }P_{x}\neq 0\\x=\frac{H}{2P_{x}}\text{, }y\in \mathrm{R}\text{, }&P_{x}\neq 0\text{ and }M=\frac{H}{2}\text{ and }P_{y}=0\end{matrix}\right.
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x=\frac{H}{2P_{x}},P_{x}x+P_{y}y=M
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.
x=\frac{H}{2P_{x}}
Pick one of the two equations which is more simple to solve for x by isolating x on the left hand side of the equal sign.
P_{x}\times \frac{H}{2P_{x}}+P_{y}y=M
Substitute \frac{H}{2P_{x}} for x in the other equation, P_{x}x+P_{y}y=M.
\frac{H}{2}+P_{y}y=M
Multiply P_{x} times \frac{H}{2P_{x}}.
P_{y}y=-\frac{H}{2}+M
Subtract \frac{H}{2} from both sides of the equation.
y=\frac{-\frac{H}{2}+M}{P_{y}}
Divide both sides by P_{y}.
x=\frac{H}{2P_{x}},y=\frac{-\frac{H}{2}+M}{P_{y}}
The system is now solved.
x=\frac{H}{2P_{x}},P_{x}x+P_{y}y=M
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.
x=\frac{H}{2P_{x}}
Pick one of the two equations which is more simple to solve for x by isolating x on the left hand side of the equal sign.
P_{x}\times \frac{H}{2P_{x}}+P_{y}y=M
Substitute \frac{H}{2P_{x}} for x in the other equation, P_{x}x+P_{y}y=M.
\frac{H}{2}+P_{y}y=M
Multiply P_{x} times \frac{H}{2P_{x}}.
P_{y}y=-\frac{H}{2}+M
Subtract \frac{H}{2} from both sides of the equation.
y=\frac{-\frac{H}{2}+M}{P_{y}}
Divide both sides by P_{y}.
x=\frac{H}{2P_{x}},y=\frac{-\frac{H}{2}+M}{P_{y}}
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}