Solve {l}{231x+269y=193}{269x+231y=307}

$$\left. \begin{array} { l } { 231 x + 269 y = 193 } \\ { 269 x + 231 y = 307 } \end{array} \right.$$

Solve for x, y

$$x=2$$

$$y=-1$$

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.

$$231x+269y=193,269x+231y=307$$

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

$$231x+269y=193$$

Subtract 269y from both sides of the equation.

$$231x=-269y+193$$

Divide both sides by 231.

$$x=\frac{1}{231}\left(-269y+193\right)$$

Multiply \frac{1}{231} times -269y+193.

$$x=-\frac{269}{231}y+\frac{193}{231}$$

Substitute \frac{-269y+193}{231} for x in the other equation, 269x+231y=307.

$$269\left(-\frac{269}{231}y+\frac{193}{231}\right)+231y=307$$

Multiply 269 times \frac{-269y+193}{231}.

$$-\frac{72361}{231}y+\frac{51917}{231}+231y=307$$

Add -\frac{72361y}{231} to 231y.

$$-\frac{19000}{231}y+\frac{51917}{231}=307$$

Subtract \frac{51917}{231} from both sides of the equation.

$$-\frac{19000}{231}y=\frac{19000}{231}$$

Divide both sides of the equation by -\frac{19000}{231}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=-1$$

Substitute -1 for y in x=-\frac{269}{231}y+\frac{193}{231}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-\frac{269}{231}\left(-1\right)+\frac{193}{231}$$

Multiply -\frac{269}{231} times -1.

$$x=\frac{269+193}{231}$$

Add \frac{193}{231} to \frac{269}{231} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=2$$

The system is now solved.

$$x=2,y=-1$$

Steps Using Matrices

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

$$231x+269y=193,269x+231y=307$$

Write the equations in matrix form.

$$\left(\begin{matrix}231&269\\269&231\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}193\\307\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}231&269\\269&231\end{matrix}\right).

$$inverse(\left(\begin{matrix}231&269\\269&231\end{matrix}\right))\left(\begin{matrix}231&269\\269&231\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}231&269\\269&231\end{matrix}\right))\left(\begin{matrix}193\\307\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}231&269\\269&231\end{matrix}\right))\left(\begin{matrix}193\\307\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}231&269\\269&231\end{matrix}\right))\left(\begin{matrix}193\\307\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{231}{231\times 231-269\times 269}&-\frac{269}{231\times 231-269\times 269}\\-\frac{269}{231\times 231-269\times 269}&\frac{231}{231\times 231-269\times 269}\end{matrix}\right)\left(\begin{matrix}193\\307\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{231}{19000}&\frac{269}{19000}\\\frac{269}{19000}&-\frac{231}{19000}\end{matrix}\right)\left(\begin{matrix}193\\307\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{231}{19000}\times 193+\frac{269}{19000}\times 307\\\frac{269}{19000}\times 193-\frac{231}{19000}\times 307\end{matrix}\right)$$

Do the arithmetic.

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

Extract the matrix elements x and y.

$$x=2,y=-1$$

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.

$$231x+269y=193,269x+231y=307$$

To make 231x and 269x equal, multiply all terms on each side of the first equation by 269 and all terms on each side of the second by 231.

$$269\times 231x+269\times 269y=269\times 193,231\times 269x+231\times 231y=231\times 307$$

Simplify.

$$62139x+72361y=51917,62139x+53361y=70917$$

Subtract 62139x+53361y=70917 from 62139x+72361y=51917 by subtracting like terms on each side of the equal sign.

$$62139x-62139x+72361y-53361y=51917-70917$$

Add 62139x to -62139x. Terms 62139x and -62139x cancel out, leaving an equation with only one variable that can be solved.

$$72361y-53361y=51917-70917$$

Add 72361y to -53361y.

$$19000y=51917-70917$$

Add 51917 to -70917.

$$19000y=-19000$$

Divide both sides by 19000.

$$y=-1$$

Substitute -1 for y in 269x+231y=307. Because the resulting equation contains only one variable, you can solve for x directly.

$$269x+231\left(-1\right)=307$$

Multiply 231 times -1.

$$269x-231=307$$

Add 231 to both sides of the equation.

$$269x=538$$

Divide both sides by 269.

$$x=2$$

The system is now solved.

$$x=2,y=-1$$