Solve {l}{40x+720y=112}{120x+2205y=340.5}

$$\left\{ \begin{array} { l } { 40 x + 720 y = 112 } \\ { 120 x + 2205 y = 340.5 } \end{array} \right.$$

Solve for x, y

$$x=1$$

$$y=0.1$$

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.

$$40x+720y=112,120x+2205y=340.5$$

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

$$40x+720y=112$$

Subtract 720y from both sides of the equation.

$$40x=-720y+112$$

Divide both sides by 40.

$$x=\frac{1}{40}\left(-720y+112\right)$$

Multiply \frac{1}{40} times -720y+112.

$$x=-18y+\frac{14}{5}$$

Substitute -18y+\frac{14}{5} for x in the other equation, 120x+2205y=340.5.

$$120\left(-18y+\frac{14}{5}\right)+2205y=340.5$$

Multiply 120 times -18y+\frac{14}{5}.

$$-2160y+336+2205y=340.5$$

Add -2160y to 2205y.

$$45y+336=340.5$$

Subtract 336 from both sides of the equation.

$$45y=4.5$$

Divide both sides by 45.

$$y=0.1$$

Substitute 0.1 for y in x=-18y+\frac{14}{5}. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-18\times 0.1+\frac{14}{5}$$

Multiply -18 times 0.1.

$$x=\frac{-9+14}{5}$$

Add \frac{14}{5} to -1.8 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.

$$x=1,y=0.1$$

Steps Using Matrices

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

$$40x+720y=112,120x+2205y=340.5$$

Write the equations in matrix form.

$$\left(\begin{matrix}40&720\\120&2205\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}112\\340.5\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}40&720\\120&2205\end{matrix}\right).

$$inverse(\left(\begin{matrix}40&720\\120&2205\end{matrix}\right))\left(\begin{matrix}40&720\\120&2205\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}40&720\\120&2205\end{matrix}\right))\left(\begin{matrix}112\\340.5\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}40&720\\120&2205\end{matrix}\right))\left(\begin{matrix}112\\340.5\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}40&720\\120&2205\end{matrix}\right))\left(\begin{matrix}112\\340.5\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2205}{40\times 2205-720\times 120}&-\frac{720}{40\times 2205-720\times 120}\\-\frac{120}{40\times 2205-720\times 120}&\frac{40}{40\times 2205-720\times 120}\end{matrix}\right)\left(\begin{matrix}112\\340.5\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{49}{40}&-\frac{2}{5}\\-\frac{1}{15}&\frac{1}{45}\end{matrix}\right)\left(\begin{matrix}112\\340.5\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{49}{40}\times 112-\frac{2}{5}\times 340.5\\-\frac{1}{15}\times 112+\frac{1}{45}\times 340.5\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1\\\frac{1}{10}\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=1,y=\frac{1}{10}$$

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.

$$40x+720y=112,120x+2205y=340.5$$

To make 40x and 120x equal, multiply all terms on each side of the first equation by 120 and all terms on each side of the second by 40.

$$120\times 40x+120\times 720y=120\times 112,40\times 120x+40\times 2205y=40\times 340.5$$

Simplify.

$$4800x+86400y=13440,4800x+88200y=13620$$

Subtract 4800x+88200y=13620 from 4800x+86400y=13440 by subtracting like terms on each side of the equal sign.

$$4800x-4800x+86400y-88200y=13440-13620$$

Add 4800x to -4800x. Terms 4800x and -4800x cancel out, leaving an equation with only one variable that can be solved.

$$86400y-88200y=13440-13620$$

Add 86400y to -88200y.

$$-1800y=13440-13620$$

Add 13440 to -13620.

$$-1800y=-180$$

Divide both sides by -1800.

$$y=\frac{1}{10}$$

Substitute \frac{1}{10} for y in 120x+2205y=340.5. Because the resulting equation contains only one variable, you can solve for x directly.

$$120x+2205\times \frac{1}{10}=340.5$$

Multiply 2205 times \frac{1}{10}.

$$120x+\frac{441}{2}=340.5$$

Subtract \frac{441}{2} from both sides of the equation.

$$120x=120$$

Divide both sides by 120.

$$x=1$$

The system is now solved.

$$x=1,y=\frac{1}{10}$$