Solve {c}{3x+2y=32}{365x+226y=265.6}

$$\left. \begin{array} { c } { 3 x + 2 y = 32 } \\ { 365 x + 226 y = 265.6 } \end{array} \right.$$

Solve for x, y

$$x = -\frac{8376}{65} = -128\frac{56}{65} \approx -128.861538462$$

$$y = \frac{13604}{65} = 209\frac{19}{65} \approx 209.292307692$$

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.

$$3x+2y=32,365x+226y=265.6$$

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

$$3x+2y=32$$

Subtract 2y from both sides of the equation.

$$3x=-2y+32$$

Divide both sides by 3.

$$x=\frac{1}{3}\left(-2y+32\right)$$

Multiply \frac{1}{3} times -2y+32.

$$x=-\frac{2}{3}y+\frac{32}{3}$$

Substitute \frac{-2y+32}{3} for x in the other equation, 365x+226y=265.6.

$$365\left(-\frac{2}{3}y+\frac{32}{3}\right)+226y=265.6$$

Multiply 365 times \frac{-2y+32}{3}.

$$-\frac{730}{3}y+\frac{11680}{3}+226y=265.6$$

Add -\frac{730y}{3} to 226y.

$$-\frac{52}{3}y+\frac{11680}{3}=265.6$$

Subtract \frac{11680}{3} from both sides of the equation.

$$-\frac{52}{3}y=-\frac{54416}{15}$$

Divide both sides of the equation by -\frac{52}{3}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=\frac{13604}{65}$$

Substitute \frac{13604}{65} for y in x=-\frac{2}{3}y+\frac{32}{3}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-\frac{2}{3}\times \frac{13604}{65}+\frac{32}{3}$$

Multiply -\frac{2}{3} times \frac{13604}{65} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{27208}{195}+\frac{32}{3}$$

Add \frac{32}{3} to -\frac{27208}{195} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{8376}{65}$$

The system is now solved.

$$x=-\frac{8376}{65},y=\frac{13604}{65}$$

Steps Using Matrices

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

$$3x+2y=32,365x+226y=265.6$$

Write the equations in matrix form.

$$\left(\begin{matrix}3&2\\365&226\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}32\\265.6\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}3&2\\365&226\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&2\\365&226\end{matrix}\right))\left(\begin{matrix}3&2\\365&226\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&2\\365&226\end{matrix}\right))\left(\begin{matrix}32\\265.6\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&2\\365&226\end{matrix}\right))\left(\begin{matrix}32\\265.6\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}3&2\\365&226\end{matrix}\right))\left(\begin{matrix}32\\265.6\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{226}{3\times 226-2\times 365}&-\frac{2}{3\times 226-2\times 365}\\-\frac{365}{3\times 226-2\times 365}&\frac{3}{3\times 226-2\times 365}\end{matrix}\right)\left(\begin{matrix}32\\265.6\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{113}{26}&\frac{1}{26}\\\frac{365}{52}&-\frac{3}{52}\end{matrix}\right)\left(\begin{matrix}32\\265.6\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{113}{26}\times 32+\frac{1}{26}\times 265.6\\\frac{365}{52}\times 32-\frac{3}{52}\times 265.6\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{8376}{65}\\\frac{13604}{65}\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=-\frac{8376}{65},y=\frac{13604}{65}$$

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.

$$3x+2y=32,365x+226y=265.6$$

To make 3x and 365x equal, multiply all terms on each side of the first equation by 365 and all terms on each side of the second by 3.

$$365\times 3x+365\times 2y=365\times 32,3\times 365x+3\times 226y=3\times 265.6$$

Simplify.

$$1095x+730y=11680,1095x+678y=796.8$$

Subtract 1095x+678y=796.8 from 1095x+730y=11680 by subtracting like terms on each side of the equal sign.

$$1095x-1095x+730y-678y=11680-796.8$$

Add 1095x to -1095x. Terms 1095x and -1095x cancel out, leaving an equation with only one variable that can be solved.

$$730y-678y=11680-796.8$$

Add 730y to -678y.

$$52y=11680-796.8$$

Add 11680 to -796.8.

$$52y=10883.2$$

Divide both sides by 52.

$$y=\frac{13604}{65}$$

Substitute \frac{13604}{65} for y in 365x+226y=265.6. Because the resulting equation contains only one variable, you can solve for x directly.

$$365x+226\times \frac{13604}{65}=265.6$$

Multiply 226 times \frac{13604}{65}.

$$365x+\frac{3074504}{65}=265.6$$

Subtract \frac{3074504}{65} from both sides of the equation.

$$365x=-\frac{611448}{13}$$

Divide both sides by 365.

$$x=-\frac{8376}{65}$$

The system is now solved.

$$x=-\frac{8376}{65},y=\frac{13604}{65}$$