Solve {l}{100x+y-100y-x=2178}{x+y=18}

$$\left\{ \begin{array} { l } { 100 x + y - 100 y - x = 2178 } \\ { x + y = 18 } \end{array} \right.$$

Solve for x, y

$$x=20$$

$$y=-2$$

Steps Using Substitution

Consider the first equation. Combine y and -100y to get -99y.

$$100x-99y-x=2178$$

Combine 100x and -x to get 99x.

$$99x-99y=2178$$

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.

$$99x-99y=2178,x+y=18$$

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

$$99x-99y=2178$$

Add 99y to both sides of the equation.

$$99x=99y+2178$$

Divide both sides by 99.

$$x=\frac{1}{99}\left(99y+2178\right)$$

Multiply \frac{1}{99} times 2178+99y.

$$x=y+22$$

Substitute y+22 for x in the other equation, x+y=18.

$$y+22+y=18$$

Add y to y.

$$2y+22=18$$

Subtract 22 from both sides of the equation.

$$2y=-4$$

Divide both sides by 2.

$$y=-2$$

Substitute -2 for y in x=y+22. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-2+22$$

Add 22 to -2.

$$x=20$$

The system is now solved.

$$x=20,y=-2$$

Steps Using Matrices

Consider the first equation. Combine y and -100y to get -99y.

$$100x-99y-x=2178$$

Combine 100x and -x to get 99x.

$$99x-99y=2178$$

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

$$99x-99y=2178,x+y=18$$

Write the equations in matrix form.

$$\left(\begin{matrix}99&-99\\1&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}2178\\18\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}99&-99\\1&1\end{matrix}\right))\left(\begin{matrix}99&-99\\1&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}99&-99\\1&1\end{matrix}\right))\left(\begin{matrix}2178\\18\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}99&-99\\1&1\end{matrix}\right))\left(\begin{matrix}2178\\18\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}99&-99\\1&1\end{matrix}\right))\left(\begin{matrix}2178\\18\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{99-\left(-99\right)}&-\frac{-99}{99-\left(-99\right)}\\-\frac{1}{99-\left(-99\right)}&\frac{99}{99-\left(-99\right)}\end{matrix}\right)\left(\begin{matrix}2178\\18\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{198}&\frac{1}{2}\\-\frac{1}{198}&\frac{1}{2}\end{matrix}\right)\left(\begin{matrix}2178\\18\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1}{198}\times 2178+\frac{1}{2}\times 18\\-\frac{1}{198}\times 2178+\frac{1}{2}\times 18\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}20\\-2\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=20,y=-2$$

Steps Using Elimination

Consider the first equation. Combine y and -100y to get -99y.

$$100x-99y-x=2178$$

Combine 100x and -x to get 99x.

$$99x-99y=2178$$

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.

$$99x-99y=2178,x+y=18$$

To make 99x and x equal, multiply all terms on each side of the first equation by 1 and all terms on each side of the second by 99.

$$99x-99y=2178,99x+99y=99\times 18$$

Simplify.

$$99x-99y=2178,99x+99y=1782$$

Subtract 99x+99y=1782 from 99x-99y=2178 by subtracting like terms on each side of the equal sign.

$$99x-99x-99y-99y=2178-1782$$

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

$$-99y-99y=2178-1782$$

Add -99y to -99y.

$$-198y=2178-1782$$

Add 2178 to -1782.

$$-198y=396$$

Divide both sides by -198.

$$y=-2$$

Substitute -2 for y in x+y=18. Because the resulting equation contains only one variable, you can solve for x directly.

$$x-2=18$$

Add 2 to both sides of the equation.

$$x=20$$

The system is now solved.

$$x=20,y=-2$$