Solve {l}{y-x=20}{14020/y=10000/x}

$$\left\{ \begin{array} { l } { y - x = 20 } \\ { \frac { 14020 } { y } = \frac { 10000 } { x } } \end{array} \right.$$

Solve for y, x

$$x = \frac{10000}{201} = 49\frac{151}{201} \approx 49.751243781$$

$$y = \frac{14020}{201} = 69\frac{151}{201} \approx 69.751243781$$

Steps Using Substitution

Consider the second equation. Variable x cannot be equal to 0 since division by zero is not defined. Variable y cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by xy, the least common multiple of y,x.

$$x\times 14020=y\times 10000$$

Subtract y\times 10000 from both sides.

$$x\times 14020-y\times 10000=0$$

Multiply -1 and 10000 to get -10000.

$$x\times 14020-10000y=0$$

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.

$$y-x=20,-10000y+14020x=0$$

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

$$y-x=20$$

Add x to both sides of the equation.

$$y=x+20$$

Substitute x+20 for y in the other equation, -10000y+14020x=0.

$$-10000\left(x+20\right)+14020x=0$$

Multiply -10000 times x+20.

$$-10000x-200000+14020x=0$$

Add -10000x to 14020x.

$$4020x-200000=0$$

Add 200000 to both sides of the equation.

$$4020x=200000$$

Divide both sides by 4020.

$$x=\frac{10000}{201}$$

Substitute \frac{10000}{201} for x in y=x+20. Because the resulting equation contains only one variable, you can solve for y directly.

$$y=\frac{10000}{201}+20$$

Add 20 to \frac{10000}{201}.

$$y=\frac{14020}{201}$$

The system is now solved.

$$y=\frac{14020}{201},x=\frac{10000}{201}$$

Steps Using Matrices

Consider the second equation. Variable x cannot be equal to 0 since division by zero is not defined. Variable y cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by xy, the least common multiple of y,x.

$$x\times 14020=y\times 10000$$

Subtract y\times 10000 from both sides.

$$x\times 14020-y\times 10000=0$$

Multiply -1 and 10000 to get -10000.

$$x\times 14020-10000y=0$$

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

$$y-x=20,-10000y+14020x=0$$

Write the equations in matrix form.

$$\left(\begin{matrix}1&-1\\-10000&14020\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}20\\0\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}1&-1\\-10000&14020\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&-1\\-10000&14020\end{matrix}\right))\left(\begin{matrix}1&-1\\-10000&14020\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\-10000&14020\end{matrix}\right))\left(\begin{matrix}20\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\-10000&14020\end{matrix}\right))\left(\begin{matrix}20\\0\end{matrix}\right)$$

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

$$\left(\begin{matrix}y\\x\end{matrix}\right)=inverse(\left(\begin{matrix}1&-1\\-10000&14020\end{matrix}\right))\left(\begin{matrix}20\\0\end{matrix}\right)$$

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.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{14020}{14020-\left(-\left(-10000\right)\right)}&-\frac{-1}{14020-\left(-\left(-10000\right)\right)}\\-\frac{-10000}{14020-\left(-\left(-10000\right)\right)}&\frac{1}{14020-\left(-\left(-10000\right)\right)}\end{matrix}\right)\left(\begin{matrix}20\\0\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{701}{201}&\frac{1}{4020}\\\frac{500}{201}&\frac{1}{4020}\end{matrix}\right)\left(\begin{matrix}20\\0\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{701}{201}\times 20\\\frac{500}{201}\times 20\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}y\\x\end{matrix}\right)=\left(\begin{matrix}\frac{14020}{201}\\\frac{10000}{201}\end{matrix}\right)$$

Extract the matrix elements y and x.

$$y=\frac{14020}{201},x=\frac{10000}{201}$$

Steps Using Elimination

Consider the second equation. Variable x cannot be equal to 0 since division by zero is not defined. Variable y cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by xy, the least common multiple of y,x.

$$x\times 14020=y\times 10000$$

Subtract y\times 10000 from both sides.

$$x\times 14020-y\times 10000=0$$

Multiply -1 and 10000 to get -10000.

$$x\times 14020-10000y=0$$

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.

$$y-x=20,-10000y+14020x=0$$

To make y and -10000y equal, multiply all terms on each side of the first equation by -10000 and all terms on each side of the second by 1.

$$-10000y-10000\left(-1\right)x=-10000\times 20,-10000y+14020x=0$$

Simplify.

$$-10000y+10000x=-200000,-10000y+14020x=0$$

Subtract -10000y+14020x=0 from -10000y+10000x=-200000 by subtracting like terms on each side of the equal sign.

$$-10000y+10000y+10000x-14020x=-200000$$

Add -10000y to 10000y. Terms -10000y and 10000y cancel out, leaving an equation with only one variable that can be solved.

$$10000x-14020x=-200000$$

Add 10000x to -14020x.

$$-4020x=-200000$$

Divide both sides by -4020.

$$x=\frac{10000}{201}$$

Substitute \frac{10000}{201} for x in -10000y+14020x=0. Because the resulting equation contains only one variable, you can solve for y directly.

$$-10000y+14020\times \frac{10000}{201}=0$$

Multiply 14020 times \frac{10000}{201}.

$$-10000y+\frac{140200000}{201}=0$$

Subtract \frac{140200000}{201} from both sides of the equation.

$$-10000y=-\frac{140200000}{201}$$

Divide both sides by -10000.

$$y=\frac{14020}{201}$$

The system is now solved.

$$y=\frac{14020}{201},x=\frac{10000}{201}$$