Solve {l}{9x-4y=2000}{7x-3y=2000}

$$\left. \begin{array} { l } { 9 x - 4 y = 2000 } \\ { 7 x - 3 y = 2000 } \end{array} \right.$$

Solve for x, y

$$x=2000$$

$$y=4000$$

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.

$$9x-4y=2000,7x-3y=2000$$

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

$$9x-4y=2000$$

Add 4y to both sides of the equation.

$$9x=4y+2000$$

Divide both sides by 9.

$$x=\frac{1}{9}\left(4y+2000\right)$$

Multiply \frac{1}{9} times 2000+4y.

$$x=\frac{4}{9}y+\frac{2000}{9}$$

Substitute \frac{2000+4y}{9} for x in the other equation, 7x-3y=2000.

$$7\left(\frac{4}{9}y+\frac{2000}{9}\right)-3y=2000$$

Multiply 7 times \frac{2000+4y}{9}.

$$\frac{28}{9}y+\frac{14000}{9}-3y=2000$$

Add \frac{28y}{9} to -3y.

$$\frac{1}{9}y+\frac{14000}{9}=2000$$

Subtract \frac{14000}{9} from both sides of the equation.

$$\frac{1}{9}y=\frac{4000}{9}$$

Multiply both sides by 9.

$$y=4000$$

Substitute 4000 for y in x=\frac{4}{9}y+\frac{2000}{9}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=\frac{4}{9}\times 4000+\frac{2000}{9}$$

Multiply \frac{4}{9} times 4000.

$$x=\frac{16000+2000}{9}$$

Add \frac{2000}{9} to \frac{16000}{9} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=2000$$

The system is now solved.

$$x=2000,y=4000$$

Steps Using Matrices

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

$$9x-4y=2000,7x-3y=2000$$

Write the equations in matrix form.

$$\left(\begin{matrix}9&-4\\7&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2000\\2000\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}9&-4\\7&-3\end{matrix}\right).

$$inverse(\left(\begin{matrix}9&-4\\7&-3\end{matrix}\right))\left(\begin{matrix}9&-4\\7&-3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}9&-4\\7&-3\end{matrix}\right))\left(\begin{matrix}2000\\2000\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}9&-4\\7&-3\end{matrix}\right))\left(\begin{matrix}2000\\2000\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}9&-4\\7&-3\end{matrix}\right))\left(\begin{matrix}2000\\2000\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{9\left(-3\right)-\left(-4\times 7\right)}&-\frac{-4}{9\left(-3\right)-\left(-4\times 7\right)}\\-\frac{7}{9\left(-3\right)-\left(-4\times 7\right)}&\frac{9}{9\left(-3\right)-\left(-4\times 7\right)}\end{matrix}\right)\left(\begin{matrix}2000\\2000\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-3&4\\-7&9\end{matrix}\right)\left(\begin{matrix}2000\\2000\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-3\times 2000+4\times 2000\\-7\times 2000+9\times 2000\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2000\\4000\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=2000,y=4000$$

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.

$$9x-4y=2000,7x-3y=2000$$

To make 9x and 7x equal, multiply all terms on each side of the first equation by 7 and all terms on each side of the second by 9.

$$7\times 9x+7\left(-4\right)y=7\times 2000,9\times 7x+9\left(-3\right)y=9\times 2000$$

Simplify.

$$63x-28y=14000,63x-27y=18000$$

Subtract 63x-27y=18000 from 63x-28y=14000 by subtracting like terms on each side of the equal sign.

$$63x-63x-28y+27y=14000-18000$$

Add 63x to -63x. Terms 63x and -63x cancel out, leaving an equation with only one variable that can be solved.

$$-28y+27y=14000-18000$$

Add -28y to 27y.

$$-y=14000-18000$$

Add 14000 to -18000.

$$-y=-4000$$

Divide both sides by -1.

$$y=4000$$

Substitute 4000 for y in 7x-3y=2000. Because the resulting equation contains only one variable, you can solve for x directly.

$$7x-3\times 4000=2000$$

Multiply -3 times 4000.

$$7x-12000=2000$$

Add 12000 to both sides of the equation.

$$7x=14000$$

Divide both sides by 7.

$$x=2000$$

The system is now solved.

$$x=2000,y=4000$$