Solve {l}{5n+2y=52000}{16n+9y=149000}

$$\left\{ \begin{array} { l } { 5 n + 2 y = 52000 } \\ { 16 n + 9 y = 149000 } \end{array} \right.$$

Solve for n, y

$$y = -\frac{87000}{13} = -6692\frac{4}{13} \approx -6692.307692308$$

$$n = \frac{170000}{13} = 13076\frac{12}{13} \approx 13076.923076923$$

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.

$$5n+2y=52000,16n+9y=149000$$

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

$$5n+2y=52000$$

Subtract 2y from both sides of the equation.

$$5n=-2y+52000$$

Divide both sides by 5.

$$n=\frac{1}{5}\left(-2y+52000\right)$$

Multiply \frac{1}{5} times -2y+52000.

$$n=-\frac{2}{5}y+10400$$

Substitute -\frac{2y}{5}+10400 for n in the other equation, 16n+9y=149000.

$$16\left(-\frac{2}{5}y+10400\right)+9y=149000$$

Multiply 16 times -\frac{2y}{5}+10400.

$$-\frac{32}{5}y+166400+9y=149000$$

Add -\frac{32y}{5} to 9y.

$$\frac{13}{5}y+166400=149000$$

Subtract 166400 from both sides of the equation.

$$\frac{13}{5}y=-17400$$

Divide both sides of the equation by \frac{13}{5}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=-\frac{87000}{13}$$

Substitute -\frac{87000}{13} for y in n=-\frac{2}{5}y+10400. Because the resulting equation contains only one variable, you can solve for n directly.

$$n=-\frac{2}{5}\left(-\frac{87000}{13}\right)+10400$$

Multiply -\frac{2}{5} times -\frac{87000}{13} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$n=\frac{34800}{13}+10400$$

Add 10400 to \frac{34800}{13}.

$$n=\frac{170000}{13}$$

The system is now solved.

$$n=\frac{170000}{13},y=-\frac{87000}{13}$$

Steps Using Matrices

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

$$5n+2y=52000,16n+9y=149000$$

Write the equations in matrix form.

$$\left(\begin{matrix}5&2\\16&9\end{matrix}\right)\left(\begin{matrix}n\\y\end{matrix}\right)=\left(\begin{matrix}52000\\149000\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}5&2\\16&9\end{matrix}\right).

$$inverse(\left(\begin{matrix}5&2\\16&9\end{matrix}\right))\left(\begin{matrix}5&2\\16&9\end{matrix}\right)\left(\begin{matrix}n\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&2\\16&9\end{matrix}\right))\left(\begin{matrix}52000\\149000\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}n\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&2\\16&9\end{matrix}\right))\left(\begin{matrix}52000\\149000\end{matrix}\right)$$

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

$$\left(\begin{matrix}n\\y\end{matrix}\right)=inverse(\left(\begin{matrix}5&2\\16&9\end{matrix}\right))\left(\begin{matrix}52000\\149000\end{matrix}\right)$$

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.

$$\left(\begin{matrix}n\\y\end{matrix}\right)=\left(\begin{matrix}\frac{9}{5\times 9-2\times 16}&-\frac{2}{5\times 9-2\times 16}\\-\frac{16}{5\times 9-2\times 16}&\frac{5}{5\times 9-2\times 16}\end{matrix}\right)\left(\begin{matrix}52000\\149000\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}n\\y\end{matrix}\right)=\left(\begin{matrix}\frac{9}{13}&-\frac{2}{13}\\-\frac{16}{13}&\frac{5}{13}\end{matrix}\right)\left(\begin{matrix}52000\\149000\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}n\\y\end{matrix}\right)=\left(\begin{matrix}\frac{9}{13}\times 52000-\frac{2}{13}\times 149000\\-\frac{16}{13}\times 52000+\frac{5}{13}\times 149000\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}n\\y\end{matrix}\right)=\left(\begin{matrix}\frac{170000}{13}\\-\frac{87000}{13}\end{matrix}\right)$$

Extract the matrix elements n and y.

$$n=\frac{170000}{13},y=-\frac{87000}{13}$$

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.

$$5n+2y=52000,16n+9y=149000$$

To make 5n and 16n equal, multiply all terms on each side of the first equation by 16 and all terms on each side of the second by 5.

$$16\times 5n+16\times 2y=16\times 52000,5\times 16n+5\times 9y=5\times 149000$$

Simplify.

$$80n+32y=832000,80n+45y=745000$$

Subtract 80n+45y=745000 from 80n+32y=832000 by subtracting like terms on each side of the equal sign.

$$80n-80n+32y-45y=832000-745000$$

Add 80n to -80n. Terms 80n and -80n cancel out, leaving an equation with only one variable that can be solved.

$$32y-45y=832000-745000$$

Add 32y to -45y.

$$-13y=832000-745000$$

Add 832000 to -745000.

$$-13y=87000$$

Divide both sides by -13.

$$y=-\frac{87000}{13}$$

Substitute -\frac{87000}{13} for y in 16n+9y=149000. Because the resulting equation contains only one variable, you can solve for n directly.

$$16n+9\left(-\frac{87000}{13}\right)=149000$$

Multiply 9 times -\frac{87000}{13}.

$$16n-\frac{783000}{13}=149000$$

Add \frac{783000}{13} to both sides of the equation.

$$16n=\frac{2720000}{13}$$

Divide both sides by 16.

$$n=\frac{170000}{13}$$

The system is now solved.

$$n=\frac{170000}{13},y=-\frac{87000}{13}$$