Solve {l}{1911(x+y)+105(x-y)=2016*190}{105(x+y)+1911(x-y)=116*201}

$$\left\{ \begin{array} { l } { 1911 ( x + y ) + 105 ( x - y ) = 2016 \times 190 } \\ { 105 ( x + y ) + 1911 ( x - y ) = 116 \times 201 } \end{array} \right.$$

Solve for x, y

$$x = \frac{33863}{336} = 100\frac{263}{336} \approx 100.782738095$$

$$y = \frac{29977}{301} = 99\frac{178}{301} \approx 99.591362126$$

Steps Using Substitution

Consider the first equation. Use the distributive property to multiply 1911 by x+y.

$$1911x+1911y+105\left(x-y\right)=2016\times 190$$

Use the distributive property to multiply 105 by x-y.

$$1911x+1911y+105x-105y=2016\times 190$$

Combine 1911x and 105x to get 2016x.

$$2016x+1911y-105y=2016\times 190$$

Combine 1911y and -105y to get 1806y.

$$2016x+1806y=2016\times 190$$

Consider the second equation. Use the distributive property to multiply 105 by x+y.

$$105x+105y+1911\left(x-y\right)=116\times 201$$

Use the distributive property to multiply 1911 by x-y.

$$105x+105y+1911x-1911y=116\times 201$$

Combine 105x and 1911x to get 2016x.

$$2016x+105y-1911y=116\times 201$$

Combine 105y and -1911y to get -1806y.

$$2016x-1806y=116\times 201$$

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.

$$2016x+1806y=383040,2016x-1806y=23316$$

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

$$2016x+1806y=383040$$

Subtract 1806y from both sides of the equation.

$$2016x=-1806y+383040$$

Divide both sides by 2016.

$$x=\frac{1}{2016}\left(-1806y+383040\right)$$

Multiply \frac{1}{2016} times -1806y+383040.

$$x=-\frac{43}{48}y+190$$

Substitute -\frac{43y}{48}+190 for x in the other equation, 2016x-1806y=23316.

$$2016\left(-\frac{43}{48}y+190\right)-1806y=23316$$

Multiply 2016 times -\frac{43y}{48}+190.

$$-1806y+383040-1806y=23316$$

Add -1806y to -1806y.

$$-3612y+383040=23316$$

Subtract 383040 from both sides of the equation.

$$-3612y=-359724$$

Divide both sides by -3612.

$$y=\frac{29977}{301}$$

Substitute \frac{29977}{301} for y in x=-\frac{43}{48}y+190. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-\frac{43}{48}\times \frac{29977}{301}+190$$

Multiply -\frac{43}{48} times \frac{29977}{301} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{29977}{336}+190$$

Add 190 to -\frac{29977}{336}.

$$x=\frac{33863}{336}$$

The system is now solved.

$$x=\frac{33863}{336},y=\frac{29977}{301}$$

Steps Using Matrices

Consider the first equation. Use the distributive property to multiply 1911 by x+y.

$$1911x+1911y+105\left(x-y\right)=2016\times 190$$

Use the distributive property to multiply 105 by x-y.

$$1911x+1911y+105x-105y=2016\times 190$$

Combine 1911x and 105x to get 2016x.

$$2016x+1911y-105y=2016\times 190$$

Combine 1911y and -105y to get 1806y.

$$2016x+1806y=2016\times 190$$

Consider the second equation. Use the distributive property to multiply 105 by x+y.

$$105x+105y+1911\left(x-y\right)=116\times 201$$

Use the distributive property to multiply 1911 by x-y.

$$105x+105y+1911x-1911y=116\times 201$$

Combine 105x and 1911x to get 2016x.

$$2016x+105y-1911y=116\times 201$$

Combine 105y and -1911y to get -1806y.

$$2016x-1806y=116\times 201$$

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

$$2016x+1806y=383040,2016x-1806y=23316$$

Write the equations in matrix form.

$$\left(\begin{matrix}2016&1806\\2016&-1806\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}383040\\23316\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}2016&1806\\2016&-1806\end{matrix}\right).

$$inverse(\left(\begin{matrix}2016&1806\\2016&-1806\end{matrix}\right))\left(\begin{matrix}2016&1806\\2016&-1806\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2016&1806\\2016&-1806\end{matrix}\right))\left(\begin{matrix}383040\\23316\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2016&1806\\2016&-1806\end{matrix}\right))\left(\begin{matrix}383040\\23316\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2016&1806\\2016&-1806\end{matrix}\right))\left(\begin{matrix}383040\\23316\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1806}{2016\left(-1806\right)-1806\times 2016}&-\frac{1806}{2016\left(-1806\right)-1806\times 2016}\\-\frac{2016}{2016\left(-1806\right)-1806\times 2016}&\frac{2016}{2016\left(-1806\right)-1806\times 2016}\end{matrix}\right)\left(\begin{matrix}383040\\23316\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{4032}&\frac{1}{4032}\\\frac{1}{3612}&-\frac{1}{3612}\end{matrix}\right)\left(\begin{matrix}383040\\23316\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{4032}\times 383040+\frac{1}{4032}\times 23316\\\frac{1}{3612}\times 383040-\frac{1}{3612}\times 23316\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{33863}{336}\\\frac{29977}{301}\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=\frac{33863}{336},y=\frac{29977}{301}$$

Steps Using Elimination

Consider the first equation. Use the distributive property to multiply 1911 by x+y.

$$1911x+1911y+105\left(x-y\right)=2016\times 190$$

Use the distributive property to multiply 105 by x-y.

$$1911x+1911y+105x-105y=2016\times 190$$

Combine 1911x and 105x to get 2016x.

$$2016x+1911y-105y=2016\times 190$$

Combine 1911y and -105y to get 1806y.

$$2016x+1806y=2016\times 190$$

Consider the second equation. Use the distributive property to multiply 105 by x+y.

$$105x+105y+1911\left(x-y\right)=116\times 201$$

Use the distributive property to multiply 1911 by x-y.

$$105x+105y+1911x-1911y=116\times 201$$

Combine 105x and 1911x to get 2016x.

$$2016x+105y-1911y=116\times 201$$

Combine 105y and -1911y to get -1806y.

$$2016x-1806y=116\times 201$$

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.

$$2016x+1806y=383040,2016x-1806y=23316$$

Subtract 2016x-1806y=23316 from 2016x+1806y=383040 by subtracting like terms on each side of the equal sign.

$$2016x-2016x+1806y+1806y=383040-23316$$

Add 2016x to -2016x. Terms 2016x and -2016x cancel out, leaving an equation with only one variable that can be solved.

$$1806y+1806y=383040-23316$$

Add 1806y to 1806y.

$$3612y=383040-23316$$

Add 383040 to -23316.

$$3612y=359724$$

Divide both sides by 3612.

$$y=\frac{29977}{301}$$

Substitute \frac{29977}{301} for y in 2016x-1806y=23316. Because the resulting equation contains only one variable, you can solve for x directly.

$$2016x-1806\times \frac{29977}{301}=23316$$

Multiply -1806 times \frac{29977}{301}.

$$2016x-179862=23316$$

Add 179862 to both sides of the equation.

$$2016x=203178$$

Divide both sides by 2016.

$$x=\frac{33863}{336}$$

The system is now solved.

$$x=\frac{33863}{336},y=\frac{29977}{301}$$