Solve {l}{217x+13ny=913}{131x+217y=827}

$$\left. \begin{array} { l } { 217 x + 13 n y = 913 } \\ { 131 x + 217 y = 827 } \end{array} \right.$$

Solve for x, y

$$x=-\frac{198121-10751n}{1703n-47089}$$

$$y=-\frac{59856}{1703n-47089}$$

$$n\neq \frac{47089}{1703}$$

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.

$$217x+13ny=913,131x+217y=827$$

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

$$217x+13ny=913$$

Subtract 13ny from both sides of the equation.

$$217x=\left(-13n\right)y+913$$

Divide both sides by 217.

$$x=\frac{1}{217}\left(\left(-13n\right)y+913\right)$$

Multiply \frac{1}{217} times -13ny+913.

$$x=\left(-\frac{13n}{217}\right)y+\frac{913}{217}$$

Substitute \frac{-13ny+913}{217} for x in the other equation, 131x+217y=827.

$$131\left(\left(-\frac{13n}{217}\right)y+\frac{913}{217}\right)+217y=827$$

Multiply 131 times \frac{-13ny+913}{217}.

$$\left(-\frac{1703n}{217}\right)y+\frac{119603}{217}+217y=827$$

Add -\frac{1703ny}{217} to 217y.

$$\left(-\frac{1703n}{217}+217\right)y+\frac{119603}{217}=827$$

Subtract \frac{119603}{217} from both sides of the equation.

$$\left(-\frac{1703n}{217}+217\right)y=\frac{59856}{217}$$

Divide both sides by -\frac{1703n}{217}+217.

$$y=\frac{59856}{47089-1703n}$$

Substitute \frac{59856}{47089-1703n} for y in x=\left(-\frac{13n}{217}\right)y+\frac{913}{217}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=\left(-\frac{13n}{217}\right)\times \frac{59856}{47089-1703n}+\frac{913}{217}$$

Multiply -\frac{13n}{217} times \frac{59856}{47089-1703n}.

$$x=-\frac{778128n}{217\left(47089-1703n\right)}+\frac{913}{217}$$

Add \frac{913}{217} to -\frac{778128n}{217\left(47089-1703n\right)}.

$$x=\frac{198121-10751n}{47089-1703n}$$

The system is now solved.

$$x=\frac{198121-10751n}{47089-1703n},y=\frac{59856}{47089-1703n}$$

Steps Using Matrices

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

$$217x+13ny=913,131x+217y=827$$

Write the equations in matrix form.

$$\left(\begin{matrix}217&13n\\131&217\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}913\\827\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}217&13n\\131&217\end{matrix}\right).

$$inverse(\left(\begin{matrix}217&13n\\131&217\end{matrix}\right))\left(\begin{matrix}217&13n\\131&217\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}217&13n\\131&217\end{matrix}\right))\left(\begin{matrix}913\\827\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}217&13n\\131&217\end{matrix}\right))\left(\begin{matrix}913\\827\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}217&13n\\131&217\end{matrix}\right))\left(\begin{matrix}913\\827\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{217}{217\times 217-13n\times 131}&-\frac{13n}{217\times 217-13n\times 131}\\-\frac{131}{217\times 217-13n\times 131}&\frac{217}{217\times 217-13n\times 131}\end{matrix}\right)\left(\begin{matrix}913\\827\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{217}{47089-1703n}&-\frac{13n}{47089-1703n}\\-\frac{131}{47089-1703n}&\frac{217}{47089-1703n}\end{matrix}\right)\left(\begin{matrix}913\\827\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{217}{47089-1703n}\times 913+\left(-\frac{13n}{47089-1703n}\right)\times 827\\\left(-\frac{131}{47089-1703n}\right)\times 913+\frac{217}{47089-1703n}\times 827\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{10751n-198121}{47089-1703n}\\\frac{59856}{47089-1703n}\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=-\frac{10751n-198121}{47089-1703n},y=\frac{59856}{47089-1703n}$$

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.

$$217x+13ny=913,131x+217y=827$$

To make 217x and 131x equal, multiply all terms on each side of the first equation by 131 and all terms on each side of the second by 217.

$$131\times 217x+131\times 13ny=131\times 913,217\times 131x+217\times 217y=217\times 827$$

Simplify.

$$28427x+1703ny=119603,28427x+47089y=179459$$

Subtract 28427x+47089y=179459 from 28427x+1703ny=119603 by subtracting like terms on each side of the equal sign.

$$28427x-28427x+1703ny-47089y=119603-179459$$

Add 28427x to -28427x. Terms 28427x and -28427x cancel out, leaving an equation with only one variable that can be solved.

$$1703ny-47089y=119603-179459$$

Add 1703ny to -47089y.

$$\left(1703n-47089\right)y=119603-179459$$

Add 119603 to -179459.

$$\left(1703n-47089\right)y=-59856$$

Divide both sides by 1703n-47089.

$$y=-\frac{59856}{1703n-47089}$$

Substitute -\frac{59856}{1703n-47089} for y in 131x+217y=827. Because the resulting equation contains only one variable, you can solve for x directly.

$$131x+217\left(-\frac{59856}{1703n-47089}\right)=827$$

Multiply 217 times -\frac{59856}{1703n-47089}.

$$131x-\frac{12988752}{1703n-47089}=827$$

Add \frac{12988752}{1703n-47089} to both sides of the equation.

$$131x=\frac{131\left(10751n-198121\right)}{1703n-47089}$$

Divide both sides by 131.

$$x=\frac{10751n-198121}{1703n-47089}$$

The system is now solved.

$$x=\frac{10751n-198121}{1703n-47089},y=-\frac{59856}{1703n-47089}$$