Solve {l}{361x+463y=-102}{463x+361y=102}

$$\left\{ \begin{array} { l } { 361 x + 463 y = - 102 } \\ { 463 x + 361 y = 102 } \end{array} \right.$$

Solve for x, y

$$x=1$$

$$y=-1$$

Steps Using Substitution

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.

$$361x+463y=-102,463x+361y=102$$

Choose one of the equations and solve it for x by isolating x on the left hand side of the equal sign.

$$361x+463y=-102$$

Subtract 463y from both sides of the equation.

$$361x=-463y-102$$

Divide both sides by 361.

$$x=\frac{1}{361}\left(-463y-102\right)$$

Multiply \frac{1}{361} times -463y-102.

$$x=-\frac{463}{361}y-\frac{102}{361}$$

Substitute \frac{-463y-102}{361} for x in the other equation, 463x+361y=102.

$$463\left(-\frac{463}{361}y-\frac{102}{361}\right)+361y=102$$

Multiply 463 times \frac{-463y-102}{361}.

$$-\frac{214369}{361}y-\frac{47226}{361}+361y=102$$

Add -\frac{214369y}{361} to 361y.

$$-\frac{84048}{361}y-\frac{47226}{361}=102$$

Add \frac{47226}{361} to both sides of the equation.

$$-\frac{84048}{361}y=\frac{84048}{361}$$

Divide both sides of the equation by -\frac{84048}{361}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=-1$$

Substitute -1 for y in x=-\frac{463}{361}y-\frac{102}{361}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-\frac{463}{361}\left(-1\right)-\frac{102}{361}$$

Multiply -\frac{463}{361} times -1.

$$x=\frac{463-102}{361}$$

Add -\frac{102}{361} to \frac{463}{361} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=1$$

The system is now solved.

$$x=1,y=-1$$

Steps Using Matrices

Put the equations in standard form and then use matrices to solve the system of equations.

$$361x+463y=-102,463x+361y=102$$

Write the equations in matrix form.

$$\left(\begin{matrix}361&463\\463&361\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-102\\102\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}361&463\\463&361\end{matrix}\right).

$$inverse(\left(\begin{matrix}361&463\\463&361\end{matrix}\right))\left(\begin{matrix}361&463\\463&361\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}361&463\\463&361\end{matrix}\right))\left(\begin{matrix}-102\\102\end{matrix}\right)$$

The product of a matrix and its inverse is the identity matrix.

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}361&463\\463&361\end{matrix}\right))\left(\begin{matrix}-102\\102\end{matrix}\right)$$

Multiply the matrices on the left hand side of the equal sign.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}361&463\\463&361\end{matrix}\right))\left(\begin{matrix}-102\\102\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{361}{361\times 361-463\times 463}&-\frac{463}{361\times 361-463\times 463}\\-\frac{463}{361\times 361-463\times 463}&\frac{361}{361\times 361-463\times 463}\end{matrix}\right)\left(\begin{matrix}-102\\102\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{361}{84048}&\frac{463}{84048}\\\frac{463}{84048}&-\frac{361}{84048}\end{matrix}\right)\left(\begin{matrix}-102\\102\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{361}{84048}\left(-102\right)+\frac{463}{84048}\times 102\\\frac{463}{84048}\left(-102\right)-\frac{361}{84048}\times 102\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1\\-1\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=1,y=-1$$

Steps Using Elimination

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.

$$361x+463y=-102,463x+361y=102$$

To make 361x and 463x equal, multiply all terms on each side of the first equation by 463 and all terms on each side of the second by 361.

$$463\times 361x+463\times 463y=463\left(-102\right),361\times 463x+361\times 361y=361\times 102$$

Simplify.

$$167143x+214369y=-47226,167143x+130321y=36822$$

Subtract 167143x+130321y=36822 from 167143x+214369y=-47226 by subtracting like terms on each side of the equal sign.

$$167143x-167143x+214369y-130321y=-47226-36822$$

Add 167143x to -167143x. Terms 167143x and -167143x cancel out, leaving an equation with only one variable that can be solved.

$$214369y-130321y=-47226-36822$$

Add 214369y to -130321y.

$$84048y=-47226-36822$$

Add -47226 to -36822.

$$84048y=-84048$$

Divide both sides by 84048.

$$y=-1$$

Substitute -1 for y in 463x+361y=102. Because the resulting equation contains only one variable, you can solve for x directly.

$$463x+361\left(-1\right)=102$$

Multiply 361 times -1.

$$463x-361=102$$

Add 361 to both sides of the equation.

$$463x=463$$

Divide both sides by 463.

$$x=1$$

The system is now solved.

$$x=1,y=-1$$