Solve {l}{-23.8501x-43.5142*(y+50)=-50y}{-23.8501(y+50)+43.5142x=50x|}

$$\left. \begin{array} { l } { - 23.8501 x - 43.5142 * ( y + 50 ) = - 50 y } \\ { - 23.8501 ( y + 50 ) + 43.5142 x = 50 x | } \end{array} \right.$$

Solve for x, y

$$x = -\frac{1192505000000}{12217857433} = -97\frac{7372828999}{12217857433} \approx -97.603446966$$

$$y = -\frac{286602871650}{12217857433} = -23\frac{5592150691}{12217857433} \approx -23.457703057$$

Steps Using Substitution

Consider the first equation. Use the distributive property to multiply -43.5142 by y+50.

$$-23.8501x-43.5142y-2175.71=-50y$$

Add 50y to both sides.

$$-23.8501x-43.5142y-2175.71+50y=0$$

Combine -43.5142y and 50y to get 6.4858y.

$$-23.8501x+6.4858y-2175.71=0$$

Add 2175.71 to both sides. Anything plus zero gives itself.

$$-23.8501x+6.4858y=2175.71$$

Consider the second equation. Use the distributive property to multiply -23.8501 by y+50.

$$-23.8501y-1192.505+43.5142x=50x$$

Subtract 50x from both sides.

$$-23.8501y-1192.505+43.5142x-50x=0$$

Combine 43.5142x and -50x to get -6.4858x.

$$-23.8501y-1192.505-6.4858x=0$$

Add 1192.505 to both sides. Anything plus zero gives itself.

$$-23.8501y-6.4858x=1192.505$$

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.

$$-23.8501x+6.4858y=2175.71,-6.4858x-23.8501y=1192.505$$

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

$$-23.8501x+6.4858y=2175.71$$

Subtract \frac{32429y}{5000} from both sides of the equation.

$$-23.8501x=-6.4858y+2175.71$$

Divide both sides of the equation by -23.8501, which is the same as multiplying both sides by the reciprocal of the fraction.

$$x=-\frac{10000}{238501}\left(-6.4858y+2175.71\right)$$

Multiply -\frac{10000}{238501} times -\frac{32429y}{5000}+2175.71.

$$x=\frac{64858}{238501}y-\frac{21757100}{238501}$$

Substitute \frac{64858y-21757100}{238501} for x in the other equation, -6.4858x-23.8501y=1192.505.

$$-6.4858\left(\frac{64858}{238501}y-\frac{21757100}{238501}\right)-23.8501y=1192.505$$

Multiply -6.4858 times \frac{64858y-21757100}{238501}.

$$-\frac{1051640041}{596252500}y+\frac{7055609959}{11925050}-23.8501y=1192.505$$

Add -\frac{1051640041y}{596252500} to -\frac{238501y}{10000}.

$$-\frac{12217857433}{477002000}y+\frac{7055609959}{11925050}=1192.505$$

Subtract \frac{7055609959}{11925050} from both sides of the equation.

$$-\frac{12217857433}{477002000}y=\frac{5732057433}{9540040}$$

Divide both sides of the equation by -\frac{12217857433}{477002000}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=-\frac{286602871650}{12217857433}$$

Substitute -\frac{286602871650}{12217857433} for y in x=\frac{64858}{238501}y-\frac{21757100}{238501}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=\frac{64858}{238501}\left(-\frac{286602871650}{12217857433}\right)-\frac{21757100}{238501}$$

Multiply \frac{64858}{238501} times -\frac{286602871650}{12217857433} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{18588489049475700}{2913971215627933}-\frac{21757100}{238501}$$

Add -\frac{21757100}{238501} to -\frac{18588489049475700}{2913971215627933} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{1192505000000}{12217857433}$$

The system is now solved.

$$x=-\frac{1192505000000}{12217857433},y=-\frac{286602871650}{12217857433}$$

Steps Using Matrices

Consider the first equation. Use the distributive property to multiply -43.5142 by y+50.

$$-23.8501x-43.5142y-2175.71=-50y$$

Add 50y to both sides.

$$-23.8501x-43.5142y-2175.71+50y=0$$

Combine -43.5142y and 50y to get 6.4858y.

$$-23.8501x+6.4858y-2175.71=0$$

Add 2175.71 to both sides. Anything plus zero gives itself.

$$-23.8501x+6.4858y=2175.71$$

Consider the second equation. Use the distributive property to multiply -23.8501 by y+50.

$$-23.8501y-1192.505+43.5142x=50x$$

Subtract 50x from both sides.

$$-23.8501y-1192.505+43.5142x-50x=0$$

Combine 43.5142x and -50x to get -6.4858x.

$$-23.8501y-1192.505-6.4858x=0$$

Add 1192.505 to both sides. Anything plus zero gives itself.

$$-23.8501y-6.4858x=1192.505$$

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

$$-23.8501x+6.4858y=2175.71,-6.4858x-23.8501y=1192.505$$

Write the equations in matrix form.

$$\left(\begin{matrix}-23.8501&6.4858\\-6.4858&-23.8501\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2175.71\\1192.505\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}-23.8501&6.4858\\-6.4858&-23.8501\end{matrix}\right).

$$inverse(\left(\begin{matrix}-23.8501&6.4858\\-6.4858&-23.8501\end{matrix}\right))\left(\begin{matrix}-23.8501&6.4858\\-6.4858&-23.8501\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-23.8501&6.4858\\-6.4858&-23.8501\end{matrix}\right))\left(\begin{matrix}2175.71\\1192.505\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-23.8501&6.4858\\-6.4858&-23.8501\end{matrix}\right))\left(\begin{matrix}2175.71\\1192.505\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-23.8501&6.4858\\-6.4858&-23.8501\end{matrix}\right))\left(\begin{matrix}2175.71\\1192.505\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{23.8501}{-23.8501\left(-23.8501\right)-6.4858\left(-6.4858\right)}&-\frac{6.4858}{-23.8501\left(-23.8501\right)-6.4858\left(-6.4858\right)}\\-\frac{-6.4858}{-23.8501\left(-23.8501\right)-6.4858\left(-6.4858\right)}&-\frac{23.8501}{-23.8501\left(-23.8501\right)-6.4858\left(-6.4858\right)}\end{matrix}\right)\left(\begin{matrix}2175.71\\1192.505\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{477002000}{12217857433}&-\frac{129716000}{12217857433}\\\frac{129716000}{12217857433}&-\frac{477002000}{12217857433}\end{matrix}\right)\left(\begin{matrix}2175.71\\1192.505\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{477002000}{12217857433}\times 2175.71-\frac{129716000}{12217857433}\times 1192.505\\\frac{129716000}{12217857433}\times 2175.71-\frac{477002000}{12217857433}\times 1192.505\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1192505000000}{12217857433}\\-\frac{286602871650}{12217857433}\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=-\frac{1192505000000}{12217857433},y=-\frac{286602871650}{12217857433}$$

Steps Using Elimination

Consider the first equation. Use the distributive property to multiply -43.5142 by y+50.

$$-23.8501x-43.5142y-2175.71=-50y$$

Add 50y to both sides.

$$-23.8501x-43.5142y-2175.71+50y=0$$

Combine -43.5142y and 50y to get 6.4858y.

$$-23.8501x+6.4858y-2175.71=0$$

Add 2175.71 to both sides. Anything plus zero gives itself.

$$-23.8501x+6.4858y=2175.71$$

Consider the second equation. Use the distributive property to multiply -23.8501 by y+50.

$$-23.8501y-1192.505+43.5142x=50x$$

Subtract 50x from both sides.

$$-23.8501y-1192.505+43.5142x-50x=0$$

Combine 43.5142x and -50x to get -6.4858x.

$$-23.8501y-1192.505-6.4858x=0$$

Add 1192.505 to both sides. Anything plus zero gives itself.

$$-23.8501y-6.4858x=1192.505$$

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.

$$-23.8501x+6.4858y=2175.71,-6.4858x-23.8501y=1192.505$$

To make -\frac{238501x}{10000} and -\frac{32429x}{5000} equal, multiply all terms on each side of the first equation by -6.4858 and all terms on each side of the second by -23.8501.

$$-6.4858\left(-23.8501\right)x-6.4858\times 6.4858y=-6.4858\times 2175.71,-23.8501\left(-6.4858\right)x-23.8501\left(-23.8501\right)y=-23.8501\times 1192.505$$

Simplify.

$$154.68697858x-42.06560164y=-14111.219918,154.68697858x+568.82727001y=-28441.3635005$$

Subtract 154.68697858x+568.82727001y=-28441.3635005 from 154.68697858x-42.06560164y=-14111.219918 by subtracting like terms on each side of the equal sign.

$$154.68697858x-154.68697858x-42.06560164y-568.82727001y=-14111.219918+28441.3635005$$

Add \frac{7734348929x}{50000000} to -\frac{7734348929x}{50000000}. Terms \frac{7734348929x}{50000000} and -\frac{7734348929x}{50000000} cancel out, leaving an equation with only one variable that can be solved.

$$-42.06560164y-568.82727001y=-14111.219918+28441.3635005$$

Add -\frac{1051640041y}{25000000} to -\frac{56882727001y}{100000000}.

$$-610.89287165y=-14111.219918+28441.3635005$$

Add -14111.219918 to 28441.3635005 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$-610.89287165y=14330.1435825$$

Divide both sides of the equation by -610.89287165, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=-\frac{286602871650}{12217857433}$$

Substitute -\frac{286602871650}{12217857433} for y in -6.4858x-23.8501y=1192.505. Because the resulting equation contains only one variable, you can solve for x directly.

$$-6.4858x-23.8501\left(-\frac{286602871650}{12217857433}\right)=1192.505$$

Multiply -23.8501 times -\frac{286602871650}{12217857433} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$-6.4858x+\frac{1367101429827933}{2443571486600}=1192.505$$

Subtract \frac{1367101429827933}{2443571486600} from both sides of the equation.

$$-6.4858x=\frac{7734348929000}{12217857433}$$

Divide both sides of the equation by -6.4858, which is the same as multiplying both sides by the reciprocal of the fraction.

$$x=-\frac{1192505000000}{12217857433}$$

The system is now solved.

$$x=-\frac{1192505000000}{12217857433},y=-\frac{286602871650}{12217857433}$$