Solve {l}{15x+9y=57000}{10x+16y=68000}

$$\left. \begin{array} { l } { 15 x + 9 y = 57000 } \\ { 10 x + 16 y = 68000 } \end{array} \right.$$

Solve for x, y

$$x=2000$$

$$y=3000$$

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.

$$15x+9y=57000,10x+16y=68000$$

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

$$15x+9y=57000$$

Subtract 9y from both sides of the equation.

$$15x=-9y+57000$$

Divide both sides by 15.

$$x=\frac{1}{15}\left(-9y+57000\right)$$

Multiply \frac{1}{15} times -9y+57000.

$$x=-\frac{3}{5}y+3800$$

Substitute -\frac{3y}{5}+3800 for x in the other equation, 10x+16y=68000.

$$10\left(-\frac{3}{5}y+3800\right)+16y=68000$$

Multiply 10 times -\frac{3y}{5}+3800.

$$-6y+38000+16y=68000$$

Add -6y to 16y.

$$10y+38000=68000$$

Subtract 38000 from both sides of the equation.

$$10y=30000$$

Divide both sides by 10.

$$y=3000$$

Substitute 3000 for y in x=-\frac{3}{5}y+3800. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-\frac{3}{5}\times 3000+3800$$

Multiply -\frac{3}{5} times 3000.

$$x=-1800+3800$$

Add 3800 to -1800.

$$x=2000$$

The system is now solved.

$$x=2000,y=3000$$

Steps Using Matrices

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

$$15x+9y=57000,10x+16y=68000$$

Write the equations in matrix form.

$$\left(\begin{matrix}15&9\\10&16\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}57000\\68000\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}15&9\\10&16\end{matrix}\right).

$$inverse(\left(\begin{matrix}15&9\\10&16\end{matrix}\right))\left(\begin{matrix}15&9\\10&16\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}15&9\\10&16\end{matrix}\right))\left(\begin{matrix}57000\\68000\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}15&9\\10&16\end{matrix}\right))\left(\begin{matrix}57000\\68000\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}15&9\\10&16\end{matrix}\right))\left(\begin{matrix}57000\\68000\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{16}{15\times 16-9\times 10}&-\frac{9}{15\times 16-9\times 10}\\-\frac{10}{15\times 16-9\times 10}&\frac{15}{15\times 16-9\times 10}\end{matrix}\right)\left(\begin{matrix}57000\\68000\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{8}{75}&-\frac{3}{50}\\-\frac{1}{15}&\frac{1}{10}\end{matrix}\right)\left(\begin{matrix}57000\\68000\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{8}{75}\times 57000-\frac{3}{50}\times 68000\\-\frac{1}{15}\times 57000+\frac{1}{10}\times 68000\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2000\\3000\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=2000,y=3000$$

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.

$$15x+9y=57000,10x+16y=68000$$

To make 15x and 10x equal, multiply all terms on each side of the first equation by 10 and all terms on each side of the second by 15.

$$10\times 15x+10\times 9y=10\times 57000,15\times 10x+15\times 16y=15\times 68000$$

Simplify.

$$150x+90y=570000,150x+240y=1020000$$

Subtract 150x+240y=1020000 from 150x+90y=570000 by subtracting like terms on each side of the equal sign.

$$150x-150x+90y-240y=570000-1020000$$

Add 150x to -150x. Terms 150x and -150x cancel out, leaving an equation with only one variable that can be solved.

$$90y-240y=570000-1020000$$

Add 90y to -240y.

$$-150y=570000-1020000$$

Add 570000 to -1020000.

$$-150y=-450000$$

Divide both sides by -150.

$$y=3000$$

Substitute 3000 for y in 10x+16y=68000. Because the resulting equation contains only one variable, you can solve for x directly.

$$10x+16\times 3000=68000$$

Multiply 16 times 3000.

$$10x+48000=68000$$

Subtract 48000 from both sides of the equation.

$$10x=20000$$

Divide both sides by 10.

$$x=2000$$

The system is now solved.

$$x=2000,y=3000$$