Solve {l}{x+y=200}{11x+9y=1960}

$$\left\{ \begin{array} { l } { x + y = 200 } \\ { 11 x + 9 y = 1960 } \end{array} \right.$$

Solve for x, y

$$x=80$$

$$y=120$$

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.

$$x+y=200,11x+9y=1960$$

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

$$x+y=200$$

Subtract y from both sides of the equation.

$$x=-y+200$$

Substitute -y+200 for x in the other equation, 11x+9y=1960.

$$11\left(-y+200\right)+9y=1960$$

Multiply 11 times -y+200.

$$-11y+2200+9y=1960$$

Add -11y to 9y.

$$-2y+2200=1960$$

Subtract 2200 from both sides of the equation.

$$-2y=-240$$

Divide both sides by -2.

$$y=120$$

Substitute 120 for y in x=-y+200. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-120+200$$

Add 200 to -120.

$$x=80$$

The system is now solved.

$$x=80,y=120$$

Steps Using Matrices

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

$$x+y=200,11x+9y=1960$$

Write the equations in matrix form.

$$\left(\begin{matrix}1&1\\11&9\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}200\\1960\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}1&1\\11&9\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&1\\11&9\end{matrix}\right))\left(\begin{matrix}1&1\\11&9\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\11&9\end{matrix}\right))\left(\begin{matrix}200\\1960\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\11&9\end{matrix}\right))\left(\begin{matrix}200\\1960\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\11&9\end{matrix}\right))\left(\begin{matrix}200\\1960\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{9}{9-11}&-\frac{1}{9-11}\\-\frac{11}{9-11}&\frac{1}{9-11}\end{matrix}\right)\left(\begin{matrix}200\\1960\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{9}{2}&\frac{1}{2}\\\frac{11}{2}&-\frac{1}{2}\end{matrix}\right)\left(\begin{matrix}200\\1960\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{9}{2}\times 200+\frac{1}{2}\times 1960\\\frac{11}{2}\times 200-\frac{1}{2}\times 1960\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}80\\120\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=80,y=120$$

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.

$$x+y=200,11x+9y=1960$$

To make x 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 1.

$$11x+11y=11\times 200,11x+9y=1960$$

Simplify.

$$11x+11y=2200,11x+9y=1960$$

Subtract 11x+9y=1960 from 11x+11y=2200 by subtracting like terms on each side of the equal sign.

$$11x-11x+11y-9y=2200-1960$$

Add 11x to -11x. Terms 11x and -11x cancel out, leaving an equation with only one variable that can be solved.

$$11y-9y=2200-1960$$

Add 11y to -9y.

$$2y=2200-1960$$

Add 2200 to -1960.

$$2y=240$$

Divide both sides by 2.

$$y=120$$

Substitute 120 for y in 11x+9y=1960. Because the resulting equation contains only one variable, you can solve for x directly.

$$11x+9\times 120=1960$$

Multiply 9 times 120.

$$11x+1080=1960$$

Subtract 1080 from both sides of the equation.

$$11x=880$$

Divide both sides by 11.

$$x=80$$

The system is now solved.

$$x=80,y=120$$