Solve {l}{13x+11y=70}{11x+13y=74}

$$\left. \begin{array} { l } { 13 x + 11 y = 70 } \\ { 11 x + 13 y = 74 } \end{array} \right.$$

Solve for x, y

$$x=2$$

$$y=4$$

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.

$$13x+11y=70,11x+13y=74$$

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

$$13x+11y=70$$

Subtract 11y from both sides of the equation.

$$13x=-11y+70$$

Divide both sides by 13.

$$x=\frac{1}{13}\left(-11y+70\right)$$

Multiply \frac{1}{13} times -11y+70.

$$x=-\frac{11}{13}y+\frac{70}{13}$$

Substitute \frac{-11y+70}{13} for x in the other equation, 11x+13y=74.

$$11\left(-\frac{11}{13}y+\frac{70}{13}\right)+13y=74$$

Multiply 11 times \frac{-11y+70}{13}.

$$-\frac{121}{13}y+\frac{770}{13}+13y=74$$

Add -\frac{121y}{13} to 13y.

$$\frac{48}{13}y+\frac{770}{13}=74$$

Subtract \frac{770}{13} from both sides of the equation.

$$\frac{48}{13}y=\frac{192}{13}$$

Divide both sides of the equation by \frac{48}{13}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=4$$

Substitute 4 for y in x=-\frac{11}{13}y+\frac{70}{13}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-\frac{11}{13}\times 4+\frac{70}{13}$$

Multiply -\frac{11}{13} times 4.

$$x=\frac{-44+70}{13}$$

Add \frac{70}{13} to -\frac{44}{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.

$$x=2,y=4$$

Steps Using Matrices

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

$$13x+11y=70,11x+13y=74$$

Write the equations in matrix form.

$$\left(\begin{matrix}13&11\\11&13\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}70\\74\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}13&11\\11&13\end{matrix}\right).

$$inverse(\left(\begin{matrix}13&11\\11&13\end{matrix}\right))\left(\begin{matrix}13&11\\11&13\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}13&11\\11&13\end{matrix}\right))\left(\begin{matrix}70\\74\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}13&11\\11&13\end{matrix}\right))\left(\begin{matrix}70\\74\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}13&11\\11&13\end{matrix}\right))\left(\begin{matrix}70\\74\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{13}{13\times 13-11\times 11}&-\frac{11}{13\times 13-11\times 11}\\-\frac{11}{13\times 13-11\times 11}&\frac{13}{13\times 13-11\times 11}\end{matrix}\right)\left(\begin{matrix}70\\74\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{13}{48}&-\frac{11}{48}\\-\frac{11}{48}&\frac{13}{48}\end{matrix}\right)\left(\begin{matrix}70\\74\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{13}{48}\times 70-\frac{11}{48}\times 74\\-\frac{11}{48}\times 70+\frac{13}{48}\times 74\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2\\4\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=2,y=4$$

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.

$$13x+11y=70,11x+13y=74$$

To make 13x and 11x equal, multiply all terms on each side of the first equation by 11 and all terms on each side of the second by 13.

$$11\times 13x+11\times 11y=11\times 70,13\times 11x+13\times 13y=13\times 74$$

Simplify.

$$143x+121y=770,143x+169y=962$$

Subtract 143x+169y=962 from 143x+121y=770 by subtracting like terms on each side of the equal sign.

$$143x-143x+121y-169y=770-962$$

Add 143x to -143x. Terms 143x and -143x cancel out, leaving an equation with only one variable that can be solved.

$$121y-169y=770-962$$

Add 121y to -169y.

$$-48y=770-962$$

Add 770 to -962.

$$-48y=-192$$

Divide both sides by -48.

$$y=4$$

Substitute 4 for y in 11x+13y=74. Because the resulting equation contains only one variable, you can solve for x directly.

$$11x+13\times 4=74$$

Multiply 13 times 4.

$$11x+52=74$$

Subtract 52 from both sides of the equation.

$$11x=22$$

Divide both sides by 11.

$$x=2$$

The system is now solved.

$$x=2,y=4$$