Solve {c}{x+y=14025}{x*1850+y*500=22000000}

$$\left. \begin{array} { c } { x + y = 14025 } \\ { x \cdot 1850 + y \cdot 500 = 22000000 } \end{array} \right.$$

Solve for x, y

$$x = \frac{299750}{27} = 11101\frac{23}{27} \approx 11101.851851852$$

$$y = \frac{78925}{27} = 2923\frac{4}{27} \approx 2923.148148148$$

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.

$$x+y=14025,1850x+500y=22000000$$

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

$$x+y=14025$$

Subtract y from both sides of the equation.

$$x=-y+14025$$

Substitute -y+14025 for x in the other equation, 1850x+500y=22000000.

$$1850\left(-y+14025\right)+500y=22000000$$

Multiply 1850 times -y+14025.

$$-1850y+25946250+500y=22000000$$

Add -1850y to 500y.

$$-1350y+25946250=22000000$$

Subtract 25946250 from both sides of the equation.

$$-1350y=-3946250$$

Divide both sides by -1350.

$$y=\frac{78925}{27}$$

Substitute \frac{78925}{27} for y in x=-y+14025. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-\frac{78925}{27}+14025$$

Add 14025 to -\frac{78925}{27}.

$$x=\frac{299750}{27}$$

The system is now solved.

$$x=\frac{299750}{27},y=\frac{78925}{27}$$

Steps Using Matrices

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

$$x+y=14025,1850x+500y=22000000$$

Write the equations in matrix form.

$$\left(\begin{matrix}1&1\\1850&500\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}14025\\22000000\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}1&1\\1850&500\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&1\\1850&500\end{matrix}\right))\left(\begin{matrix}1&1\\1850&500\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\1850&500\end{matrix}\right))\left(\begin{matrix}14025\\22000000\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\1850&500\end{matrix}\right))\left(\begin{matrix}14025\\22000000\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\1850&500\end{matrix}\right))\left(\begin{matrix}14025\\22000000\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{500}{500-1850}&-\frac{1}{500-1850}\\-\frac{1850}{500-1850}&\frac{1}{500-1850}\end{matrix}\right)\left(\begin{matrix}14025\\22000000\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{10}{27}&\frac{1}{1350}\\\frac{37}{27}&-\frac{1}{1350}\end{matrix}\right)\left(\begin{matrix}14025\\22000000\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{10}{27}\times 14025+\frac{1}{1350}\times 22000000\\\frac{37}{27}\times 14025-\frac{1}{1350}\times 22000000\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{299750}{27}\\\frac{78925}{27}\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=\frac{299750}{27},y=\frac{78925}{27}$$

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.

$$x+y=14025,1850x+500y=22000000$$

To make x and 1850x equal, multiply all terms on each side of the first equation by 1850 and all terms on each side of the second by 1.

$$1850x+1850y=1850\times 14025,1850x+500y=22000000$$

Simplify.

$$1850x+1850y=25946250,1850x+500y=22000000$$

Subtract 1850x+500y=22000000 from 1850x+1850y=25946250 by subtracting like terms on each side of the equal sign.

$$1850x-1850x+1850y-500y=25946250-22000000$$

Add 1850x to -1850x. Terms 1850x and -1850x cancel out, leaving an equation with only one variable that can be solved.

$$1850y-500y=25946250-22000000$$

Add 1850y to -500y.

$$1350y=25946250-22000000$$

Add 25946250 to -22000000.

$$1350y=3946250$$

Divide both sides by 1350.

$$y=\frac{78925}{27}$$

Substitute \frac{78925}{27} for y in 1850x+500y=22000000. Because the resulting equation contains only one variable, you can solve for x directly.

$$1850x+500\times \frac{78925}{27}=22000000$$

Multiply 500 times \frac{78925}{27}.

$$1850x+\frac{39462500}{27}=22000000$$

Subtract \frac{39462500}{27} from both sides of the equation.

$$1850x=\frac{554537500}{27}$$

Divide both sides by 1850.

$$x=\frac{299750}{27}$$

The system is now solved.

$$x=\frac{299750}{27},y=\frac{78925}{27}$$