Solve {l}{x+y=104}{13x+11y=1240}

$$\left. \begin{array} { l } { x + y = 104 } \\ { 13 x + 11 y = 1240 } \end{array} \right.$$

Solve for x, y

$$x=48$$

$$y=56$$

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.

$$x+y=104,13x+11y=1240$$

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

$$x+y=104$$

Subtract y from both sides of the equation.

$$x=-y+104$$

Substitute -y+104 for x in the other equation, 13x+11y=1240.

$$13\left(-y+104\right)+11y=1240$$

Multiply 13 times -y+104.

$$-13y+1352+11y=1240$$

Add -13y to 11y.

$$-2y+1352=1240$$

Subtract 1352 from both sides of the equation.

$$-2y=-112$$

Divide both sides by -2.

$$y=56$$

Substitute 56 for y in x=-y+104. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-56+104$$

Add 104 to -56.

$$x=48$$

The system is now solved.

$$x=48,y=56$$

Steps Using Matrices

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

$$x+y=104,13x+11y=1240$$

Write the equations in matrix form.

$$\left(\begin{matrix}1&1\\13&11\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}104\\1240\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}1&1\\13&11\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&1\\13&11\end{matrix}\right))\left(\begin{matrix}1&1\\13&11\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\13&11\end{matrix}\right))\left(\begin{matrix}104\\1240\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\13&11\end{matrix}\right))\left(\begin{matrix}104\\1240\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\13&11\end{matrix}\right))\left(\begin{matrix}104\\1240\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{11}{11-13}&-\frac{1}{11-13}\\-\frac{13}{11-13}&\frac{1}{11-13}\end{matrix}\right)\left(\begin{matrix}104\\1240\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{11}{2}&\frac{1}{2}\\\frac{13}{2}&-\frac{1}{2}\end{matrix}\right)\left(\begin{matrix}104\\1240\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{11}{2}\times 104+\frac{1}{2}\times 1240\\\frac{13}{2}\times 104-\frac{1}{2}\times 1240\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}48\\56\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=48,y=56$$

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.

$$x+y=104,13x+11y=1240$$

To make x and 13x equal, multiply all terms on each side of the first equation by 13 and all terms on each side of the second by 1.

$$13x+13y=13\times 104,13x+11y=1240$$

Simplify.

$$13x+13y=1352,13x+11y=1240$$

Subtract 13x+11y=1240 from 13x+13y=1352 by subtracting like terms on each side of the equal sign.

$$13x-13x+13y-11y=1352-1240$$

Add 13x to -13x. Terms 13x and -13x cancel out, leaving an equation with only one variable that can be solved.

$$13y-11y=1352-1240$$

Add 13y to -11y.

$$2y=1352-1240$$

Add 1352 to -1240.

$$2y=112$$

Divide both sides by 2.

$$y=56$$

Substitute 56 for y in 13x+11y=1240. Because the resulting equation contains only one variable, you can solve for x directly.

$$13x+11\times 56=1240$$

Multiply 11 times 56.

$$13x+616=1240$$

Subtract 616 from both sides of the equation.

$$13x=624$$

Divide both sides by 13.

$$x=48$$

The system is now solved.

$$x=48,y=56$$