Solve {l}{71x+37y=253}{37x+71y=287}

$$\left. \begin{array} { l } { 71 x + 37 y = 253 } \\ { 37 x + 71 y = 287 } \end{array} \right.$$

Solve for x, y

$$x=2$$

$$y=3$$

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.

$$71x+37y=253,37x+71y=287$$

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

$$71x+37y=253$$

Subtract 37y from both sides of the equation.

$$71x=-37y+253$$

Divide both sides by 71.

$$x=\frac{1}{71}\left(-37y+253\right)$$

Multiply \frac{1}{71} times -37y+253.

$$x=-\frac{37}{71}y+\frac{253}{71}$$

Substitute \frac{-37y+253}{71} for x in the other equation, 37x+71y=287.

$$37\left(-\frac{37}{71}y+\frac{253}{71}\right)+71y=287$$

Multiply 37 times \frac{-37y+253}{71}.

$$-\frac{1369}{71}y+\frac{9361}{71}+71y=287$$

Add -\frac{1369y}{71} to 71y.

$$\frac{3672}{71}y+\frac{9361}{71}=287$$

Subtract \frac{9361}{71} from both sides of the equation.

$$\frac{3672}{71}y=\frac{11016}{71}$$

Divide both sides of the equation by \frac{3672}{71}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=3$$

Substitute 3 for y in x=-\frac{37}{71}y+\frac{253}{71}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-\frac{37}{71}\times 3+\frac{253}{71}$$

Multiply -\frac{37}{71} times 3.

$$x=\frac{-111+253}{71}$$

Add \frac{253}{71} to -\frac{111}{71} 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.

$$x=2,y=3$$

Steps Using Matrices

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

$$71x+37y=253,37x+71y=287$$

Write the equations in matrix form.

$$\left(\begin{matrix}71&37\\37&71\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}253\\287\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}71&37\\37&71\end{matrix}\right).

$$inverse(\left(\begin{matrix}71&37\\37&71\end{matrix}\right))\left(\begin{matrix}71&37\\37&71\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}71&37\\37&71\end{matrix}\right))\left(\begin{matrix}253\\287\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}71&37\\37&71\end{matrix}\right))\left(\begin{matrix}253\\287\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}71&37\\37&71\end{matrix}\right))\left(\begin{matrix}253\\287\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{71}{71\times 71-37\times 37}&-\frac{37}{71\times 71-37\times 37}\\-\frac{37}{71\times 71-37\times 37}&\frac{71}{71\times 71-37\times 37}\end{matrix}\right)\left(\begin{matrix}253\\287\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{71}{3672}&-\frac{37}{3672}\\-\frac{37}{3672}&\frac{71}{3672}\end{matrix}\right)\left(\begin{matrix}253\\287\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{71}{3672}\times 253-\frac{37}{3672}\times 287\\-\frac{37}{3672}\times 253+\frac{71}{3672}\times 287\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2\\3\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=2,y=3$$

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.

$$71x+37y=253,37x+71y=287$$

To make 71x and 37x equal, multiply all terms on each side of the first equation by 37 and all terms on each side of the second by 71.

$$37\times 71x+37\times 37y=37\times 253,71\times 37x+71\times 71y=71\times 287$$

Simplify.

$$2627x+1369y=9361,2627x+5041y=20377$$

Subtract 2627x+5041y=20377 from 2627x+1369y=9361 by subtracting like terms on each side of the equal sign.

$$2627x-2627x+1369y-5041y=9361-20377$$

Add 2627x to -2627x. Terms 2627x and -2627x cancel out, leaving an equation with only one variable that can be solved.

$$1369y-5041y=9361-20377$$

Add 1369y to -5041y.

$$-3672y=9361-20377$$

Add 9361 to -20377.

$$-3672y=-11016$$

Divide both sides by -3672.

$$y=3$$

Substitute 3 for y in 37x+71y=287. Because the resulting equation contains only one variable, you can solve for x directly.

$$37x+71\times 3=287$$

Multiply 71 times 3.

$$37x+213=287$$

Subtract 213 from both sides of the equation.

$$37x=74$$

Divide both sides by 37.

$$x=2$$

The system is now solved.

$$x=2,y=3$$