Solve {l}{x+y=25}{6000x+4600y=108000}

$$\left\{ \begin{array} { l } { x + y = 25 } \\ { 6000 x + 4600 y = 108000 } \end{array} \right.$$

Solve for x, y

$$x=-5$$

$$y=30$$

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.

$$x+y=25,6000x+4600y=108000$$

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

$$x+y=25$$

Subtract y from both sides of the equation.

$$x=-y+25$$

Substitute -y+25 for x in the other equation, 6000x+4600y=108000.

$$6000\left(-y+25\right)+4600y=108000$$

Multiply 6000 times -y+25.

$$-6000y+150000+4600y=108000$$

Add -6000y to 4600y.

$$-1400y+150000=108000$$

Subtract 150000 from both sides of the equation.

$$-1400y=-42000$$

Divide both sides by -1400.

$$y=30$$

Substitute 30 for y in x=-y+25. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-30+25$$

Add 25 to -30.

$$x=-5$$

The system is now solved.

$$x=-5,y=30$$

Steps Using Matrices

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

$$x+y=25,6000x+4600y=108000$$

Write the equations in matrix form.

$$\left(\begin{matrix}1&1\\6000&4600\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}25\\108000\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}1&1\\6000&4600\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&1\\6000&4600\end{matrix}\right))\left(\begin{matrix}1&1\\6000&4600\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\6000&4600\end{matrix}\right))\left(\begin{matrix}25\\108000\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\6000&4600\end{matrix}\right))\left(\begin{matrix}25\\108000\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\6000&4600\end{matrix}\right))\left(\begin{matrix}25\\108000\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4600}{4600-6000}&-\frac{1}{4600-6000}\\-\frac{6000}{4600-6000}&\frac{1}{4600-6000}\end{matrix}\right)\left(\begin{matrix}25\\108000\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{23}{7}&\frac{1}{1400}\\\frac{30}{7}&-\frac{1}{1400}\end{matrix}\right)\left(\begin{matrix}25\\108000\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{23}{7}\times 25+\frac{1}{1400}\times 108000\\\frac{30}{7}\times 25-\frac{1}{1400}\times 108000\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-5\\30\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=-5,y=30$$

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.

$$x+y=25,6000x+4600y=108000$$

To make x and 6000x equal, multiply all terms on each side of the first equation by 6000 and all terms on each side of the second by 1.

$$6000x+6000y=6000\times 25,6000x+4600y=108000$$

Simplify.

$$6000x+6000y=150000,6000x+4600y=108000$$

Subtract 6000x+4600y=108000 from 6000x+6000y=150000 by subtracting like terms on each side of the equal sign.

$$6000x-6000x+6000y-4600y=150000-108000$$

Add 6000x to -6000x. Terms 6000x and -6000x cancel out, leaving an equation with only one variable that can be solved.

$$6000y-4600y=150000-108000$$

Add 6000y to -4600y.

$$1400y=150000-108000$$

Add 150000 to -108000.

$$1400y=42000$$

Divide both sides by 1400.

$$y=30$$

Substitute 30 for y in 6000x+4600y=108000. Because the resulting equation contains only one variable, you can solve for x directly.

$$6000x+4600\times 30=108000$$

Multiply 4600 times 30.

$$6000x+138000=108000$$

Subtract 138000 from both sides of the equation.

$$6000x=-30000$$

Divide both sides by 6000.

$$x=-5$$

The system is now solved.

$$x=-5,y=30$$