Solve {l}{800=340/340-xf}{400=340/340+xf}

$$\left\{ \begin{array} { l } { 800 = \frac { 340 } { 340 - x } f } \\ { 400 = \frac { 340 } { 340 + x } f } \end{array} \right.$$

Solve for x, f

$$x = \frac{340}{3} = 113\frac{1}{3} \approx 113.333333333$$

$$f = \frac{1600}{3} = 533\frac{1}{3} \approx 533.333333333$$

Steps Using Substitution

Consider the first equation. Variable x cannot be equal to 340 since division by zero is not defined. Multiply both sides of the equation by -x+340.

$$800\left(-x+340\right)=340f$$

Use the distributive property to multiply 800 by -x+340.

$$-800x+272000=340f$$

Subtract 340f from both sides.

$$-800x+272000-340f=0$$

Subtract 272000 from both sides. Anything subtracted from zero gives its negation.

$$-800x-340f=-272000$$

Consider the second equation. Variable x cannot be equal to -340 since division by zero is not defined. Multiply both sides of the equation by x+340.

$$400\left(x+340\right)=340f$$

Use the distributive property to multiply 400 by x+340.

$$400x+136000=340f$$

Subtract 340f from both sides.

$$400x+136000-340f=0$$

Subtract 136000 from both sides. Anything subtracted from zero gives its negation.

$$400x-340f=-136000$$

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.

$$-800x-340f=-272000,400x-340f=-136000$$

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

$$-800x-340f=-272000$$

Add 340f to both sides of the equation.

$$-800x=340f-272000$$

Divide both sides by -800.

$$x=-\frac{1}{800}\left(340f-272000\right)$$

Multiply -\frac{1}{800} times -272000+340f.

$$x=-\frac{17}{40}f+340$$

Substitute -\frac{17f}{40}+340 for x in the other equation, 400x-340f=-136000.

$$400\left(-\frac{17}{40}f+340\right)-340f=-136000$$

Multiply 400 times -\frac{17f}{40}+340.

$$-170f+136000-340f=-136000$$

Add -170f to -340f.

$$-510f+136000=-136000$$

Subtract 136000 from both sides of the equation.

$$-510f=-272000$$

Divide both sides by -510.

$$f=\frac{1600}{3}$$

Substitute \frac{1600}{3} for f in x=-\frac{17}{40}f+340. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-\frac{17}{40}\times \frac{1600}{3}+340$$

Multiply -\frac{17}{40} times \frac{1600}{3} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{680}{3}+340$$

Add 340 to -\frac{680}{3}.

$$x=\frac{340}{3}$$

The system is now solved.

$$x=\frac{340}{3},f=\frac{1600}{3}$$

Steps Using Matrices

Consider the first equation. Variable x cannot be equal to 340 since division by zero is not defined. Multiply both sides of the equation by -x+340.

$$800\left(-x+340\right)=340f$$

Use the distributive property to multiply 800 by -x+340.

$$-800x+272000=340f$$

Subtract 340f from both sides.

$$-800x+272000-340f=0$$

Subtract 272000 from both sides. Anything subtracted from zero gives its negation.

$$-800x-340f=-272000$$

Consider the second equation. Variable x cannot be equal to -340 since division by zero is not defined. Multiply both sides of the equation by x+340.

$$400\left(x+340\right)=340f$$

Use the distributive property to multiply 400 by x+340.

$$400x+136000=340f$$

Subtract 340f from both sides.

$$400x+136000-340f=0$$

Subtract 136000 from both sides. Anything subtracted from zero gives its negation.

$$400x-340f=-136000$$

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

$$-800x-340f=-272000,400x-340f=-136000$$

Write the equations in matrix form.

$$\left(\begin{matrix}-800&-340\\400&-340\end{matrix}\right)\left(\begin{matrix}x\\f\end{matrix}\right)=\left(\begin{matrix}-272000\\-136000\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}-800&-340\\400&-340\end{matrix}\right).

$$inverse(\left(\begin{matrix}-800&-340\\400&-340\end{matrix}\right))\left(\begin{matrix}-800&-340\\400&-340\end{matrix}\right)\left(\begin{matrix}x\\f\end{matrix}\right)=inverse(\left(\begin{matrix}-800&-340\\400&-340\end{matrix}\right))\left(\begin{matrix}-272000\\-136000\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\f\end{matrix}\right)=inverse(\left(\begin{matrix}-800&-340\\400&-340\end{matrix}\right))\left(\begin{matrix}-272000\\-136000\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\f\end{matrix}\right)=inverse(\left(\begin{matrix}-800&-340\\400&-340\end{matrix}\right))\left(\begin{matrix}-272000\\-136000\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\f\end{matrix}\right)=\left(\begin{matrix}-\frac{340}{-800\left(-340\right)-\left(-340\times 400\right)}&-\frac{-340}{-800\left(-340\right)-\left(-340\times 400\right)}\\-\frac{400}{-800\left(-340\right)-\left(-340\times 400\right)}&-\frac{800}{-800\left(-340\right)-\left(-340\times 400\right)}\end{matrix}\right)\left(\begin{matrix}-272000\\-136000\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\f\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{1200}&\frac{1}{1200}\\-\frac{1}{1020}&-\frac{1}{510}\end{matrix}\right)\left(\begin{matrix}-272000\\-136000\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\f\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{1200}\left(-272000\right)+\frac{1}{1200}\left(-136000\right)\\-\frac{1}{1020}\left(-272000\right)-\frac{1}{510}\left(-136000\right)\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\f\end{matrix}\right)=\left(\begin{matrix}\frac{340}{3}\\\frac{1600}{3}\end{matrix}\right)$$

Extract the matrix elements x and f.

$$x=\frac{340}{3},f=\frac{1600}{3}$$

Steps Using Elimination

Consider the first equation. Variable x cannot be equal to 340 since division by zero is not defined. Multiply both sides of the equation by -x+340.

$$800\left(-x+340\right)=340f$$

Use the distributive property to multiply 800 by -x+340.

$$-800x+272000=340f$$

Subtract 340f from both sides.

$$-800x+272000-340f=0$$

Subtract 272000 from both sides. Anything subtracted from zero gives its negation.

$$-800x-340f=-272000$$

Consider the second equation. Variable x cannot be equal to -340 since division by zero is not defined. Multiply both sides of the equation by x+340.

$$400\left(x+340\right)=340f$$

Use the distributive property to multiply 400 by x+340.

$$400x+136000=340f$$

Subtract 340f from both sides.

$$400x+136000-340f=0$$

Subtract 136000 from both sides. Anything subtracted from zero gives its negation.

$$400x-340f=-136000$$

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.

$$-800x-340f=-272000,400x-340f=-136000$$

Subtract 400x-340f=-136000 from -800x-340f=-272000 by subtracting like terms on each side of the equal sign.

$$-800x-400x-340f+340f=-272000+136000$$

Add -340f to 340f. Terms -340f and 340f cancel out, leaving an equation with only one variable that can be solved.

$$-800x-400x=-272000+136000$$

Add -800x to -400x.

$$-1200x=-272000+136000$$

Add -272000 to 136000.

$$-1200x=-136000$$

Divide both sides by -1200.

$$x=\frac{340}{3}$$

Substitute \frac{340}{3} for x in 400x-340f=-136000. Because the resulting equation contains only one variable, you can solve for f directly.

$$400\times \frac{340}{3}-340f=-136000$$

Multiply 400 times \frac{340}{3}.

$$\frac{136000}{3}-340f=-136000$$

Subtract \frac{136000}{3} from both sides of the equation.

$$-340f=-\frac{544000}{3}$$

Divide both sides by -340.

$$f=\frac{1600}{3}$$

The system is now solved.

$$x=\frac{340}{3},f=\frac{1600}{3}$$