Solve {c}{13x+650+11(y-5)=1240}{x+y=104}

$$\left\{ \begin{array} { c } { 13 x + 650 + 11 ( y - 5 ) = 1240 } \\ { x + y = 104 } \end{array} \right.$$

Solve for x, y

$$x = -\frac{499}{2} = -249\frac{1}{2} = -249.5$$

$$y = \frac{707}{2} = 353\frac{1}{2} = 353.5$$

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.

$$13x+11\left(y-5\right)+650=1240,x+y=104$$

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

$$13x+11\left(y-5\right)+650=1240$$

Multiply 11 times y-5.

$$13x+11y-55+650=1240$$

Add -55 to 650.

$$13x+11y+595=1240$$

Subtract 595 from both sides of the equation.

$$13x+11y=645$$

Subtract 11y from both sides of the equation.

$$13x=-11y+645$$

Divide both sides by 13.

$$x=\frac{1}{13}\left(-11y+645\right)$$

Multiply \frac{1}{13} times -11y+645.

$$x=-\frac{11}{13}y+\frac{645}{13}$$

Substitute \frac{-11y+645}{13} for x in the other equation, x+y=104.

$$-\frac{11}{13}y+\frac{645}{13}+y=104$$

Add -\frac{11y}{13} to y.

$$\frac{2}{13}y+\frac{645}{13}=104$$

Subtract \frac{645}{13} from both sides of the equation.

$$\frac{2}{13}y=\frac{707}{13}$$

Divide both sides of the equation by \frac{2}{13}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=\frac{707}{2}$$

Substitute \frac{707}{2} for y in x=-\frac{11}{13}y+\frac{645}{13}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-\frac{11}{13}\times \frac{707}{2}+\frac{645}{13}$$

Multiply -\frac{11}{13} times \frac{707}{2} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{7777}{26}+\frac{645}{13}$$

Add \frac{645}{13} to -\frac{7777}{26} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{499}{2}$$

The system is now solved.

$$x=-\frac{499}{2},y=\frac{707}{2}$$

Steps Using Matrices

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

$$13x+11\left(y-5\right)+650=1240,x+y=104$$

Simplify the first equation to put it in standard form.

$$13x+11\left(y-5\right)+650=1240$$

Multiply 11 times y-5.

$$13x+11y-55+650=1240$$

Add -55 to 650.

$$13x+11y+595=1240$$

Subtract 595 from both sides of the equation.

$$13x+11y=645$$

Write the equations in matrix form.

$$\left(\begin{matrix}13&11\\1&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}645\\104\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}13&11\\1&1\end{matrix}\right))\left(\begin{matrix}13&11\\1&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}13&11\\1&1\end{matrix}\right))\left(\begin{matrix}645\\104\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}13&11\\1&1\end{matrix}\right))\left(\begin{matrix}645\\104\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}13&11\\1&1\end{matrix}\right))\left(\begin{matrix}645\\104\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{13-11}&-\frac{11}{13-11}\\-\frac{1}{13-11}&\frac{13}{13-11}\end{matrix}\right)\left(\begin{matrix}645\\104\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}&-\frac{11}{2}\\-\frac{1}{2}&\frac{13}{2}\end{matrix}\right)\left(\begin{matrix}645\\104\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}\times 645-\frac{11}{2}\times 104\\-\frac{1}{2}\times 645+\frac{13}{2}\times 104\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{499}{2}\\\frac{707}{2}\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=-\frac{499}{2},y=\frac{707}{2}$$