Solve {l}{x+y=50}{36x=16y}

$$\left. \begin{array} { l } { x + y = 50 } \\ { 36 x = 16 y } \end{array} \right.$$

Solve for x, y

$$x = \frac{200}{13} = 15\frac{5}{13} \approx 15.384615385$$

$$y = \frac{450}{13} = 34\frac{8}{13} \approx 34.615384615$$

Steps Using Substitution

Consider the second equation. Subtract 16y from both sides.

$$36x-16y=0$$

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.

$$x+y=50,36x-16y=0$$

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

$$x+y=50$$

Subtract y from both sides of the equation.

$$x=-y+50$$

Substitute -y+50 for x in the other equation, 36x-16y=0.

$$36\left(-y+50\right)-16y=0$$

Multiply 36 times -y+50.

$$-36y+1800-16y=0$$

Add -36y to -16y.

$$-52y+1800=0$$

Subtract 1800 from both sides of the equation.

$$-52y=-1800$$

Divide both sides by -52.

$$y=\frac{450}{13}$$

Substitute \frac{450}{13} for y in x=-y+50. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-\frac{450}{13}+50$$

Add 50 to -\frac{450}{13}.

$$x=\frac{200}{13}$$

The system is now solved.

$$x=\frac{200}{13},y=\frac{450}{13}$$

Steps Using Matrices

Consider the second equation. Subtract 16y from both sides.

$$36x-16y=0$$

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

$$x+y=50,36x-16y=0$$

Write the equations in matrix form.

$$\left(\begin{matrix}1&1\\36&-16\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}50\\0\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}1&1\\36&-16\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&1\\36&-16\end{matrix}\right))\left(\begin{matrix}1&1\\36&-16\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\36&-16\end{matrix}\right))\left(\begin{matrix}50\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\36&-16\end{matrix}\right))\left(\begin{matrix}50\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\36&-16\end{matrix}\right))\left(\begin{matrix}50\\0\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{16}{-16-36}&-\frac{1}{-16-36}\\-\frac{36}{-16-36}&\frac{1}{-16-36}\end{matrix}\right)\left(\begin{matrix}50\\0\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{13}&\frac{1}{52}\\\frac{9}{13}&-\frac{1}{52}\end{matrix}\right)\left(\begin{matrix}50\\0\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{13}\times 50\\\frac{9}{13}\times 50\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{200}{13}\\\frac{450}{13}\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=\frac{200}{13},y=\frac{450}{13}$$

Steps Using Elimination

Consider the second equation. Subtract 16y from both sides.

$$36x-16y=0$$

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.

$$x+y=50,36x-16y=0$$

To make x and 36x equal, multiply all terms on each side of the first equation by 36 and all terms on each side of the second by 1.

$$36x+36y=36\times 50,36x-16y=0$$

Simplify.

$$36x+36y=1800,36x-16y=0$$

Subtract 36x-16y=0 from 36x+36y=1800 by subtracting like terms on each side of the equal sign.

$$36x-36x+36y+16y=1800$$

Add 36x to -36x. Terms 36x and -36x cancel out, leaving an equation with only one variable that can be solved.

$$36y+16y=1800$$

Add 36y to 16y.

$$52y=1800$$

Divide both sides by 52.

$$y=\frac{450}{13}$$

Substitute \frac{450}{13} for y in 36x-16y=0. Because the resulting equation contains only one variable, you can solve for x directly.

$$36x-16\times \frac{450}{13}=0$$

Multiply -16 times \frac{450}{13}.

$$36x-\frac{7200}{13}=0$$

Add \frac{7200}{13} to both sides of the equation.

$$36x=\frac{7200}{13}$$

Divide both sides by 36.

$$x=\frac{200}{13}$$

The system is now solved.

$$x=\frac{200}{13},y=\frac{450}{13}$$