Solve {l}{139x+361y=861}{361x+139y=639}

$$\left\{ \begin{array} { l } { 139 x + 361 y = 861 } \\ { 361 x + 139 y = 639 } \end{array} \right.$$

Solve for x, y

$$x=1$$

$$y=2$$

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.

$$139x+361y=861,361x+139y=639$$

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

$$139x+361y=861$$

Subtract 361y from both sides of the equation.

$$139x=-361y+861$$

Divide both sides by 139.

$$x=\frac{1}{139}\left(-361y+861\right)$$

Multiply \frac{1}{139} times -361y+861.

$$x=-\frac{361}{139}y+\frac{861}{139}$$

Substitute \frac{-361y+861}{139} for x in the other equation, 361x+139y=639.

$$361\left(-\frac{361}{139}y+\frac{861}{139}\right)+139y=639$$

Multiply 361 times \frac{-361y+861}{139}.

$$-\frac{130321}{139}y+\frac{310821}{139}+139y=639$$

Add -\frac{130321y}{139} to 139y.

$$-\frac{111000}{139}y+\frac{310821}{139}=639$$

Subtract \frac{310821}{139} from both sides of the equation.

$$-\frac{111000}{139}y=-\frac{222000}{139}$$

Divide both sides of the equation by -\frac{111000}{139}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=2$$

Substitute 2 for y in x=-\frac{361}{139}y+\frac{861}{139}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-\frac{361}{139}\times 2+\frac{861}{139}$$

Multiply -\frac{361}{139} times 2.

$$x=\frac{-722+861}{139}$$

Add \frac{861}{139} to -\frac{722}{139} 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.

$$x=1,y=2$$

Steps Using Matrices

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

$$139x+361y=861,361x+139y=639$$

Write the equations in matrix form.

$$\left(\begin{matrix}139&361\\361&139\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}861\\639\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}139&361\\361&139\end{matrix}\right).

$$inverse(\left(\begin{matrix}139&361\\361&139\end{matrix}\right))\left(\begin{matrix}139&361\\361&139\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}139&361\\361&139\end{matrix}\right))\left(\begin{matrix}861\\639\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}139&361\\361&139\end{matrix}\right))\left(\begin{matrix}861\\639\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}139&361\\361&139\end{matrix}\right))\left(\begin{matrix}861\\639\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{139}{139\times 139-361\times 361}&-\frac{361}{139\times 139-361\times 361}\\-\frac{361}{139\times 139-361\times 361}&\frac{139}{139\times 139-361\times 361}\end{matrix}\right)\left(\begin{matrix}861\\639\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{139}{111000}&\frac{361}{111000}\\\frac{361}{111000}&-\frac{139}{111000}\end{matrix}\right)\left(\begin{matrix}861\\639\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{139}{111000}\times 861+\frac{361}{111000}\times 639\\\frac{361}{111000}\times 861-\frac{139}{111000}\times 639\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1\\2\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=1,y=2$$

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.

$$139x+361y=861,361x+139y=639$$

To make 139x and 361x equal, multiply all terms on each side of the first equation by 361 and all terms on each side of the second by 139.

$$361\times 139x+361\times 361y=361\times 861,139\times 361x+139\times 139y=139\times 639$$

Simplify.

$$50179x+130321y=310821,50179x+19321y=88821$$

Subtract 50179x+19321y=88821 from 50179x+130321y=310821 by subtracting like terms on each side of the equal sign.

$$50179x-50179x+130321y-19321y=310821-88821$$

Add 50179x to -50179x. Terms 50179x and -50179x cancel out, leaving an equation with only one variable that can be solved.

$$130321y-19321y=310821-88821$$

Add 130321y to -19321y.

$$111000y=310821-88821$$

Add 310821 to -88821.

$$111000y=222000$$

Divide both sides by 111000.

$$y=2$$

Substitute 2 for y in 361x+139y=639. Because the resulting equation contains only one variable, you can solve for x directly.

$$361x+139\times 2=639$$

Multiply 139 times 2.

$$361x+278=639$$

Subtract 278 from both sides of the equation.

$$361x=361$$

Divide both sides by 361.

$$x=1$$

The system is now solved.

$$x=1,y=2$$