Solve {l}{x+y=68}{99x-99y=2178}

$$\left\{ \begin{array} { l } { x + y = 68 } \\ { 99 x - 99 y = 2178 } \end{array} \right.$$

Solve for x, y

$$x=45$$

$$y=23$$

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.

$$x+y=68,99x-99y=2178$$

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

$$x+y=68$$

Subtract y from both sides of the equation.

$$x=-y+68$$

Substitute -y+68 for x in the other equation, 99x-99y=2178.

$$99\left(-y+68\right)-99y=2178$$

Multiply 99 times -y+68.

$$-99y+6732-99y=2178$$

Add -99y to -99y.

$$-198y+6732=2178$$

Subtract 6732 from both sides of the equation.

$$-198y=-4554$$

Divide both sides by -198.

$$y=23$$

Substitute 23 for y in x=-y+68. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-23+68$$

Add 68 to -23.

$$x=45$$

The system is now solved.

$$x=45,y=23$$

Steps Using Matrices

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

$$x+y=68,99x-99y=2178$$

Write the equations in matrix form.

$$\left(\begin{matrix}1&1\\99&-99\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}68\\2178\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}1&1\\99&-99\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&1\\99&-99\end{matrix}\right))\left(\begin{matrix}1&1\\99&-99\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\99&-99\end{matrix}\right))\left(\begin{matrix}68\\2178\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\99&-99\end{matrix}\right))\left(\begin{matrix}68\\2178\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\99&-99\end{matrix}\right))\left(\begin{matrix}68\\2178\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{99}{-99-99}&-\frac{1}{-99-99}\\-\frac{99}{-99-99}&\frac{1}{-99-99}\end{matrix}\right)\left(\begin{matrix}68\\2178\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}&\frac{1}{198}\\\frac{1}{2}&-\frac{1}{198}\end{matrix}\right)\left(\begin{matrix}68\\2178\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{2}\times 68+\frac{1}{198}\times 2178\\\frac{1}{2}\times 68-\frac{1}{198}\times 2178\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}45\\23\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=45,y=23$$

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.

$$x+y=68,99x-99y=2178$$

To make x 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 1.

$$99x+99y=99\times 68,99x-99y=2178$$

Simplify.

$$99x+99y=6732,99x-99y=2178$$

Subtract 99x-99y=2178 from 99x+99y=6732 by subtracting like terms on each side of the equal sign.

$$99x-99x+99y+99y=6732-2178$$

Add 99x to -99x. Terms 99x and -99x cancel out, leaving an equation with only one variable that can be solved.

$$99y+99y=6732-2178$$

Add 99y to 99y.

$$198y=6732-2178$$

Add 6732 to -2178.

$$198y=4554$$

Divide both sides by 198.

$$y=23$$

Substitute 23 for y in 99x-99y=2178. Because the resulting equation contains only one variable, you can solve for x directly.

$$99x-99\times 23=2178$$

Multiply -99 times 23.

$$99x-2277=2178$$

Add 2277 to both sides of the equation.

$$99x=4455$$

Divide both sides by 99.

$$x=45$$

The system is now solved.

$$x=45,y=23$$