Solve {l}{157x+101y=56}{213x+269y=-56}

$$\left\{ \begin{array} { l } { 157 x + 101 y = 56 } \\ { 213 x + 269 y = - 56 } \end{array} \right.$$

Solve for x, y

$$x=1$$

$$y=-1$$

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.

$$157x+101y=56,213x+269y=-56$$

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

$$157x+101y=56$$

Subtract 101y from both sides of the equation.

$$157x=-101y+56$$

Divide both sides by 157.

$$x=\frac{1}{157}\left(-101y+56\right)$$

Multiply \frac{1}{157} times -101y+56.

$$x=-\frac{101}{157}y+\frac{56}{157}$$

Substitute \frac{-101y+56}{157} for x in the other equation, 213x+269y=-56.

$$213\left(-\frac{101}{157}y+\frac{56}{157}\right)+269y=-56$$

Multiply 213 times \frac{-101y+56}{157}.

$$-\frac{21513}{157}y+\frac{11928}{157}+269y=-56$$

Add -\frac{21513y}{157} to 269y.

$$\frac{20720}{157}y+\frac{11928}{157}=-56$$

Subtract \frac{11928}{157} from both sides of the equation.

$$\frac{20720}{157}y=-\frac{20720}{157}$$

Divide both sides of the equation by \frac{20720}{157}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=-1$$

Substitute -1 for y in x=-\frac{101}{157}y+\frac{56}{157}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-\frac{101}{157}\left(-1\right)+\frac{56}{157}$$

Multiply -\frac{101}{157} times -1.

$$x=\frac{101+56}{157}$$

Add \frac{56}{157} to \frac{101}{157} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=1$$

The system is now solved.

$$x=1,y=-1$$

Steps Using Matrices

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

$$157x+101y=56,213x+269y=-56$$

Write the equations in matrix form.

$$\left(\begin{matrix}157&101\\213&269\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}56\\-56\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}157&101\\213&269\end{matrix}\right).

$$inverse(\left(\begin{matrix}157&101\\213&269\end{matrix}\right))\left(\begin{matrix}157&101\\213&269\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}157&101\\213&269\end{matrix}\right))\left(\begin{matrix}56\\-56\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}157&101\\213&269\end{matrix}\right))\left(\begin{matrix}56\\-56\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}157&101\\213&269\end{matrix}\right))\left(\begin{matrix}56\\-56\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{269}{157\times 269-101\times 213}&-\frac{101}{157\times 269-101\times 213}\\-\frac{213}{157\times 269-101\times 213}&\frac{157}{157\times 269-101\times 213}\end{matrix}\right)\left(\begin{matrix}56\\-56\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{269}{20720}&-\frac{101}{20720}\\-\frac{213}{20720}&\frac{157}{20720}\end{matrix}\right)\left(\begin{matrix}56\\-56\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{269}{20720}\times 56-\frac{101}{20720}\left(-56\right)\\-\frac{213}{20720}\times 56+\frac{157}{20720}\left(-56\right)\end{matrix}\right)$$

Do the arithmetic.

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

Extract the matrix elements x and y.

$$x=1,y=-1$$

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.

$$157x+101y=56,213x+269y=-56$$

To make 157x and 213x equal, multiply all terms on each side of the first equation by 213 and all terms on each side of the second by 157.

$$213\times 157x+213\times 101y=213\times 56,157\times 213x+157\times 269y=157\left(-56\right)$$

Simplify.

$$33441x+21513y=11928,33441x+42233y=-8792$$

Subtract 33441x+42233y=-8792 from 33441x+21513y=11928 by subtracting like terms on each side of the equal sign.

$$33441x-33441x+21513y-42233y=11928+8792$$

Add 33441x to -33441x. Terms 33441x and -33441x cancel out, leaving an equation with only one variable that can be solved.

$$21513y-42233y=11928+8792$$

Add 21513y to -42233y.

$$-20720y=11928+8792$$

Add 11928 to 8792.

$$-20720y=20720$$

Divide both sides by -20720.

$$y=-1$$

Substitute -1 for y in 213x+269y=-56. Because the resulting equation contains only one variable, you can solve for x directly.

$$213x+269\left(-1\right)=-56$$

Multiply 269 times -1.

$$213x-269=-56$$

Add 269 to both sides of the equation.

$$213x=213$$

Divide both sides by 213.

$$x=1$$

The system is now solved.

$$x=1,y=-1$$