Solve {l}{101x-99y=6}{99x-101y=-6}

$$\left. \begin{array} { l } { 101 x - 99 y = 6 } \\ { 99 x - 101 y = - 6 } \end{array} \right.$$

Solve for x, y

$$x=3$$

$$y=3$$

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.

$$101x-99y=6,99x-101y=-6$$

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

$$101x-99y=6$$

Add 99y to both sides of the equation.

$$101x=99y+6$$

Divide both sides by 101.

$$x=\frac{1}{101}\left(99y+6\right)$$

Multiply \frac{1}{101} times 99y+6.

$$x=\frac{99}{101}y+\frac{6}{101}$$

Substitute \frac{99y+6}{101} for x in the other equation, 99x-101y=-6.

$$99\left(\frac{99}{101}y+\frac{6}{101}\right)-101y=-6$$

Multiply 99 times \frac{99y+6}{101}.

$$\frac{9801}{101}y+\frac{594}{101}-101y=-6$$

Add \frac{9801y}{101} to -101y.

$$-\frac{400}{101}y+\frac{594}{101}=-6$$

Subtract \frac{594}{101} from both sides of the equation.

$$-\frac{400}{101}y=-\frac{1200}{101}$$

Divide both sides of the equation by -\frac{400}{101}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=3$$

Substitute 3 for y in x=\frac{99}{101}y+\frac{6}{101}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=\frac{99}{101}\times 3+\frac{6}{101}$$

Multiply \frac{99}{101} times 3.

$$x=\frac{297+6}{101}$$

Add \frac{6}{101} to \frac{297}{101} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=3$$

The system is now solved.

$$x=3,y=3$$

Steps Using Matrices

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

$$101x-99y=6,99x-101y=-6$$

Write the equations in matrix form.

$$\left(\begin{matrix}101&-99\\99&-101\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}6\\-6\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}101&-99\\99&-101\end{matrix}\right).

$$inverse(\left(\begin{matrix}101&-99\\99&-101\end{matrix}\right))\left(\begin{matrix}101&-99\\99&-101\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}101&-99\\99&-101\end{matrix}\right))\left(\begin{matrix}6\\-6\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}101&-99\\99&-101\end{matrix}\right))\left(\begin{matrix}6\\-6\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}101&-99\\99&-101\end{matrix}\right))\left(\begin{matrix}6\\-6\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{101}{101\left(-101\right)-\left(-99\times 99\right)}&-\frac{-99}{101\left(-101\right)-\left(-99\times 99\right)}\\-\frac{99}{101\left(-101\right)-\left(-99\times 99\right)}&\frac{101}{101\left(-101\right)-\left(-99\times 99\right)}\end{matrix}\right)\left(\begin{matrix}6\\-6\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{101}{400}&-\frac{99}{400}\\\frac{99}{400}&-\frac{101}{400}\end{matrix}\right)\left(\begin{matrix}6\\-6\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{101}{400}\times 6-\frac{99}{400}\left(-6\right)\\\frac{99}{400}\times 6-\frac{101}{400}\left(-6\right)\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}3\\3\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=3,y=3$$

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.

$$101x-99y=6,99x-101y=-6$$

To make 101x and 99x equal, multiply all terms on each side of the first equation by 99 and all terms on each side of the second by 101.

$$99\times 101x+99\left(-99\right)y=99\times 6,101\times 99x+101\left(-101\right)y=101\left(-6\right)$$

Simplify.

$$9999x-9801y=594,9999x-10201y=-606$$

Subtract 9999x-10201y=-606 from 9999x-9801y=594 by subtracting like terms on each side of the equal sign.

$$9999x-9999x-9801y+10201y=594+606$$

Add 9999x to -9999x. Terms 9999x and -9999x cancel out, leaving an equation with only one variable that can be solved.

$$-9801y+10201y=594+606$$

Add -9801y to 10201y.

$$400y=594+606$$

Add 594 to 606.

$$400y=1200$$

Divide both sides by 400.

$$y=3$$

Substitute 3 for y in 99x-101y=-6. Because the resulting equation contains only one variable, you can solve for x directly.

$$99x-101\times 3=-6$$

Multiply -101 times 3.

$$99x-303=-6$$

Add 303 to both sides of the equation.

$$99x=297$$

Divide both sides by 99.

$$x=3$$

The system is now solved.

$$x=3,y=3$$