Solve {l}{x/36-y=21}{x-y/20=16}

$$\left. \begin{array} { l } { \frac { x } { 36 } - y = 21 } \\ { x - \frac { y } { 20 } = 16 } \end{array} \right.$$

Solve for x, y

$$x = \frac{10764}{719} = 14\frac{698}{719} \approx 14.970792768$$

$$y = -\frac{14800}{719} = -20\frac{420}{719} \approx -20.584144645$$

Steps Using Substitution

Consider the first equation. Multiply both sides of the equation by 36.

$$x-36y=756$$

Consider the second equation. Multiply both sides of the equation by 20.

$$20x-y=320$$

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.

$$x-36y=756,20x-y=320$$

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

$$x-36y=756$$

Add 36y to both sides of the equation.

$$x=36y+756$$

Substitute 756+36y for x in the other equation, 20x-y=320.

$$20\left(36y+756\right)-y=320$$

Multiply 20 times 756+36y.

$$720y+15120-y=320$$

Add 720y to -y.

$$719y+15120=320$$

Subtract 15120 from both sides of the equation.

$$719y=-14800$$

Divide both sides by 719.

$$y=-\frac{14800}{719}$$

Substitute -\frac{14800}{719} for y in x=36y+756. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=36\left(-\frac{14800}{719}\right)+756$$

Multiply 36 times -\frac{14800}{719}.

$$x=-\frac{532800}{719}+756$$

Add 756 to -\frac{532800}{719}.

$$x=\frac{10764}{719}$$

The system is now solved.

$$x=\frac{10764}{719},y=-\frac{14800}{719}$$

Steps Using Matrices

Consider the first equation. Multiply both sides of the equation by 36.

$$x-36y=756$$

Consider the second equation. Multiply both sides of the equation by 20.

$$20x-y=320$$

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

$$x-36y=756,20x-y=320$$

Write the equations in matrix form.

$$\left(\begin{matrix}1&-36\\20&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}756\\320\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}1&-36\\20&-1\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&-36\\20&-1\end{matrix}\right))\left(\begin{matrix}1&-36\\20&-1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-36\\20&-1\end{matrix}\right))\left(\begin{matrix}756\\320\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-36\\20&-1\end{matrix}\right))\left(\begin{matrix}756\\320\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&-36\\20&-1\end{matrix}\right))\left(\begin{matrix}756\\320\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{-1-\left(-36\times 20\right)}&-\frac{-36}{-1-\left(-36\times 20\right)}\\-\frac{20}{-1-\left(-36\times 20\right)}&\frac{1}{-1-\left(-36\times 20\right)}\end{matrix}\right)\left(\begin{matrix}756\\320\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{719}&\frac{36}{719}\\-\frac{20}{719}&\frac{1}{719}\end{matrix}\right)\left(\begin{matrix}756\\320\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{719}\times 756+\frac{36}{719}\times 320\\-\frac{20}{719}\times 756+\frac{1}{719}\times 320\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{10764}{719}\\-\frac{14800}{719}\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=\frac{10764}{719},y=-\frac{14800}{719}$$

Steps Using Elimination

Consider the first equation. Multiply both sides of the equation by 36.

$$x-36y=756$$

Consider the second equation. Multiply both sides of the equation by 20.

$$20x-y=320$$

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.

$$x-36y=756,20x-y=320$$

To make x and 20x equal, multiply all terms on each side of the first equation by 20 and all terms on each side of the second by 1.

$$20x+20\left(-36\right)y=20\times 756,20x-y=320$$

Simplify.

$$20x-720y=15120,20x-y=320$$

Subtract 20x-y=320 from 20x-720y=15120 by subtracting like terms on each side of the equal sign.

$$20x-20x-720y+y=15120-320$$

Add 20x to -20x. Terms 20x and -20x cancel out, leaving an equation with only one variable that can be solved.

$$-720y+y=15120-320$$

Add -720y to y.

$$-719y=15120-320$$

Add 15120 to -320.

$$-719y=14800$$

Divide both sides by -719.

$$y=-\frac{14800}{719}$$

Substitute -\frac{14800}{719} for y in 20x-y=320. Because the resulting equation contains only one variable, you can solve for x directly.

$$20x-\left(-\frac{14800}{719}\right)=320$$

Subtract \frac{14800}{719} from both sides of the equation.

$$20x=\frac{215280}{719}$$

Divide both sides by 20.

$$x=\frac{10764}{719}$$

The system is now solved.

$$x=\frac{10764}{719},y=-\frac{14800}{719}$$