Solve {l}{33x-y=123}{99x+7y=129}

$$\left\{ \begin{array} { l } { 33 x - y = 123 } \\ { 99 x + 7 y = 129 } \end{array} \right.$$

Solve for x, y

$$x=3$$

$$y=-24$$

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.

$$33x-y=123,99x+7y=129$$

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

$$33x-y=123$$

Add y to both sides of the equation.

$$33x=y+123$$

Divide both sides by 33.

$$x=\frac{1}{33}\left(y+123\right)$$

Multiply \frac{1}{33} times y+123.

$$x=\frac{1}{33}y+\frac{41}{11}$$

Substitute \frac{41}{11}+\frac{y}{33} for x in the other equation, 99x+7y=129.

$$99\left(\frac{1}{33}y+\frac{41}{11}\right)+7y=129$$

Multiply 99 times \frac{41}{11}+\frac{y}{33}.

$$3y+369+7y=129$$

Add 3y to 7y.

$$10y+369=129$$

Subtract 369 from both sides of the equation.

$$10y=-240$$

Divide both sides by 10.

$$y=-24$$

Substitute -24 for y in x=\frac{1}{33}y+\frac{41}{11}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=\frac{1}{33}\left(-24\right)+\frac{41}{11}$$

Multiply \frac{1}{33} times -24.

$$x=\frac{-8+41}{11}$$

Add \frac{41}{11} to -\frac{8}{11} 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.

$$x=3,y=-24$$

Steps Using Matrices

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

$$33x-y=123,99x+7y=129$$

Write the equations in matrix form.

$$\left(\begin{matrix}33&-1\\99&7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}123\\129\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}33&-1\\99&7\end{matrix}\right).

$$inverse(\left(\begin{matrix}33&-1\\99&7\end{matrix}\right))\left(\begin{matrix}33&-1\\99&7\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}33&-1\\99&7\end{matrix}\right))\left(\begin{matrix}123\\129\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}33&-1\\99&7\end{matrix}\right))\left(\begin{matrix}123\\129\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}33&-1\\99&7\end{matrix}\right))\left(\begin{matrix}123\\129\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{33\times 7-\left(-99\right)}&-\frac{-1}{33\times 7-\left(-99\right)}\\-\frac{99}{33\times 7-\left(-99\right)}&\frac{33}{33\times 7-\left(-99\right)}\end{matrix}\right)\left(\begin{matrix}123\\129\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{330}&\frac{1}{330}\\-\frac{3}{10}&\frac{1}{10}\end{matrix}\right)\left(\begin{matrix}123\\129\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{7}{330}\times 123+\frac{1}{330}\times 129\\-\frac{3}{10}\times 123+\frac{1}{10}\times 129\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}3\\-24\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=3,y=-24$$

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.

$$33x-y=123,99x+7y=129$$

To make 33x 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 33.

$$99\times 33x+99\left(-1\right)y=99\times 123,33\times 99x+33\times 7y=33\times 129$$

Simplify.

$$3267x-99y=12177,3267x+231y=4257$$

Subtract 3267x+231y=4257 from 3267x-99y=12177 by subtracting like terms on each side of the equal sign.

$$3267x-3267x-99y-231y=12177-4257$$

Add 3267x to -3267x. Terms 3267x and -3267x cancel out, leaving an equation with only one variable that can be solved.

$$-99y-231y=12177-4257$$

Add -99y to -231y.

$$-330y=12177-4257$$

Add 12177 to -4257.

$$-330y=7920$$

Divide both sides by -330.

$$y=-24$$

Substitute -24 for y in 99x+7y=129. Because the resulting equation contains only one variable, you can solve for x directly.

$$99x+7\left(-24\right)=129$$

Multiply 7 times -24.

$$99x-168=129$$

Add 168 to both sides of the equation.

$$99x=297$$

Divide both sides by 99.

$$x=3$$

The system is now solved.

$$x=3,y=-24$$