Solve {l}{13x+14y=40}{13y+14x=41}

$$\left\{ \begin{array} { l } { 13 x + 14 y = 40 } \\ { 13 y + 14 x = 41 } \end{array} \right.$$

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.

$$13x+14y=40,14x+13y=41$$

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

$$13x+14y=40$$

Subtract 14y from both sides of the equation.

$$13x=-14y+40$$

Divide both sides by 13.

$$x=\frac{1}{13}\left(-14y+40\right)$$

Multiply \frac{1}{13} times -14y+40.

$$x=-\frac{14}{13}y+\frac{40}{13}$$

Substitute \frac{-14y+40}{13} for x in the other equation, 14x+13y=41.

$$14\left(-\frac{14}{13}y+\frac{40}{13}\right)+13y=41$$

Multiply 14 times \frac{-14y+40}{13}.

$$-\frac{196}{13}y+\frac{560}{13}+13y=41$$

Add -\frac{196y}{13} to 13y.

$$-\frac{27}{13}y+\frac{560}{13}=41$$

Subtract \frac{560}{13} from both sides of the equation.

$$-\frac{27}{13}y=-\frac{27}{13}$$

Divide both sides of the equation by -\frac{27}{13}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=1$$

Substitute 1 for y in x=-\frac{14}{13}y+\frac{40}{13}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=\frac{-14+40}{13}$$

Add \frac{40}{13} to -\frac{14}{13} 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.

$$13x+14y=40,14x+13y=41$$

Write the equations in matrix form.

$$\left(\begin{matrix}13&14\\14&13\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}40\\41\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}13&14\\14&13\end{matrix}\right).

$$inverse(\left(\begin{matrix}13&14\\14&13\end{matrix}\right))\left(\begin{matrix}13&14\\14&13\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}13&14\\14&13\end{matrix}\right))\left(\begin{matrix}40\\41\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}13&14\\14&13\end{matrix}\right))\left(\begin{matrix}40\\41\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}13&14\\14&13\end{matrix}\right))\left(\begin{matrix}40\\41\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{13}{13\times 13-14\times 14}&-\frac{14}{13\times 13-14\times 14}\\-\frac{14}{13\times 13-14\times 14}&\frac{13}{13\times 13-14\times 14}\end{matrix}\right)\left(\begin{matrix}40\\41\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{13}{27}&\frac{14}{27}\\\frac{14}{27}&-\frac{13}{27}\end{matrix}\right)\left(\begin{matrix}40\\41\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{13}{27}\times 40+\frac{14}{27}\times 41\\\frac{14}{27}\times 40-\frac{13}{27}\times 41\end{matrix}\right)$$

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.

$$13x+14y=40,14x+13y=41$$

To make 13x and 14x equal, multiply all terms on each side of the first equation by 14 and all terms on each side of the second by 13.

$$14\times 13x+14\times 14y=14\times 40,13\times 14x+13\times 13y=13\times 41$$

Simplify.

$$182x+196y=560,182x+169y=533$$

Subtract 182x+169y=533 from 182x+196y=560 by subtracting like terms on each side of the equal sign.

$$182x-182x+196y-169y=560-533$$

Add 182x to -182x. Terms 182x and -182x cancel out, leaving an equation with only one variable that can be solved.

$$196y-169y=560-533$$

Add 196y to -169y.

$$27y=560-533$$

Add 560 to -533.

$$27y=27$$

Divide both sides by 27.

$$y=1$$

Substitute 1 for y in 14x+13y=41. Because the resulting equation contains only one variable, you can solve for x directly.

$$14x+13=41$$

Subtract 13 from both sides of the equation.

$$14x=28$$

Divide both sides by 14.

$$x=2$$

The system is now solved.

$$x=2,y=1$$