Solve for y, x
x=0
y=0
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y=10x
Consider the first equation. Combine 4x and 6x to get 10x.
10x-56x=0
Substitute 10x for y in the other equation, y-56x=0.
-46x=0
Add 10x to -56x.
x=0
Divide both sides by -46.
y=0
Substitute 0 for x in y=10x. Because the resulting equation contains only one variable, you can solve for y directly.
y=0,x=0
The system is now solved.
y=10x
Consider the first equation. Combine 4x and 6x to get 10x.
y-10x=0
Subtract 10x from both sides.
y=56x
Consider the second equation. Multiply 7 and 8 to get 56.
y-56x=0
Subtract 56x from both sides.
y-10x=0,y-56x=0
Put the equations in standard form and then use matrices to solve the system of equations.
\left(\begin{matrix}1&-10\\1&-56\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}0\\0\end{matrix}\right)
Write the equations in matrix form.
inverse(\left(\begin{matrix}1&-10\\1&-56\end{matrix}\right))\left(\begin{matrix}1&-10\\1&-56\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-10\\1&-56\end{matrix}\right))\left(\begin{matrix}0\\0\end{matrix}\right)
Left multiply the equation by the inverse matrix of \left(\begin{matrix}1&-10\\1&-56\end{matrix}\right).
\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-10\\1&-56\end{matrix}\right))\left(\begin{matrix}0\\0\end{matrix}\right)
The product of a matrix and its inverse is the identity matrix.
\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-10\\1&-56\end{matrix}\right))\left(\begin{matrix}0\\0\end{matrix}\right)
Multiply the matrices on the left hand side of the equal sign.
\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}-\frac{56}{-56-\left(-10\right)}&-\frac{-10}{-56-\left(-10\right)}\\-\frac{1}{-56-\left(-10\right)}&\frac{1}{-56-\left(-10\right)}\end{matrix}\right)\left(\begin{matrix}0\\0\end{matrix}\right)
For the 2\times 2 matrix \left(\begin{matrix}a&b\\c&d\end{matrix}\right), the inverse matrix is \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right), so the matrix equation can be rewritten as a matrix multiplication problem.
\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{28}{23}&-\frac{5}{23}\\\frac{1}{46}&-\frac{1}{46}\end{matrix}\right)\left(\begin{matrix}0\\0\end{matrix}\right)
Do the arithmetic.
\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}0\\0\end{matrix}\right)
Multiply the matrices.
y=0,x=0
Extract the matrix elements y and x.
y=10x
Consider the first equation. Combine 4x and 6x to get 10x.
y-10x=0
Subtract 10x from both sides.
y=56x
Consider the second equation. Multiply 7 and 8 to get 56.
y-56x=0
Subtract 56x from both sides.
y-10x=0,y-56x=0
In order to solve by elimination, coefficients of one of the variables must be the same in both equations so that the variable will cancel out when one equation is subtracted from the other.
y-y-10x+56x=0
Subtract y-56x=0 from y-10x=0 by subtracting like terms on each side of the equal sign.
-10x+56x=0
Add y to -y. Terms y and -y cancel out, leaving an equation with only one variable that can be solved.
46x=0
Add -10x to 56x.
x=0
Divide both sides by 46.
y=0
Substitute 0 for x in y-56x=0. Because the resulting equation contains only one variable, you can solve for y directly.
y=0,x=0
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}