Solve {l}{2x+3y=380}{4x+3y=360}

$$\left\{ \begin{array} { l } { 2 x + 3 y = 380 } \\ { 4 x + 3 y = 360 } \end{array} \right.$$

Solve for x, y

$$x=-10$$

$$y = \frac{400}{3} = 133\frac{1}{3} \approx 133.333333333$$

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.

$$2x+3y=380,4x+3y=360$$

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

$$2x+3y=380$$

Subtract 3y from both sides of the equation.

$$2x=-3y+380$$

Divide both sides by 2.

$$x=\frac{1}{2}\left(-3y+380\right)$$

Multiply \frac{1}{2} times -3y+380.

$$x=-\frac{3}{2}y+190$$

Substitute -\frac{3y}{2}+190 for x in the other equation, 4x+3y=360.

$$4\left(-\frac{3}{2}y+190\right)+3y=360$$

Multiply 4 times -\frac{3y}{2}+190.

$$-6y+760+3y=360$$

Add -6y to 3y.

$$-3y+760=360$$

Subtract 760 from both sides of the equation.

$$-3y=-400$$

Divide both sides by -3.

$$y=\frac{400}{3}$$

Substitute \frac{400}{3} for y in x=-\frac{3}{2}y+190. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-\frac{3}{2}\times \frac{400}{3}+190$$

Multiply -\frac{3}{2} times \frac{400}{3} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=-200+190$$

Add 190 to -200.

$$x=-10$$

The system is now solved.

$$x=-10,y=\frac{400}{3}$$

Steps Using Matrices

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

$$2x+3y=380,4x+3y=360$$

Write the equations in matrix form.

$$\left(\begin{matrix}2&3\\4&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}380\\360\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}2&3\\4&3\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&3\\4&3\end{matrix}\right))\left(\begin{matrix}2&3\\4&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\4&3\end{matrix}\right))\left(\begin{matrix}380\\360\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\4&3\end{matrix}\right))\left(\begin{matrix}380\\360\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\4&3\end{matrix}\right))\left(\begin{matrix}380\\360\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{2\times 3-3\times 4}&-\frac{3}{2\times 3-3\times 4}\\-\frac{4}{2\times 3-3\times 4}&\frac{2}{2\times 3-3\times 4}\end{matrix}\right)\left(\begin{matrix}380\\360\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{2}&\frac{1}{2}\\\frac{2}{3}&-\frac{1}{3}\end{matrix}\right)\left(\begin{matrix}380\\360\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{2}\times 380+\frac{1}{2}\times 360\\\frac{2}{3}\times 380-\frac{1}{3}\times 360\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-10\\\frac{400}{3}\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=-10,y=\frac{400}{3}$$

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.

$$2x+3y=380,4x+3y=360$$

Subtract 4x+3y=360 from 2x+3y=380 by subtracting like terms on each side of the equal sign.

$$2x-4x+3y-3y=380-360$$

Add 3y to -3y. Terms 3y and -3y cancel out, leaving an equation with only one variable that can be solved.

$$2x-4x=380-360$$

Add 2x to -4x.

$$-2x=380-360$$

Add 380 to -360.

$$-2x=20$$

Divide both sides by -2.

$$x=-10$$

Substitute -10 for x in 4x+3y=360. Because the resulting equation contains only one variable, you can solve for y directly.

$$4\left(-10\right)+3y=360$$

Multiply 4 times -10.

$$-40+3y=360$$

Add 40 to both sides of the equation.

$$3y=400$$

Divide both sides by 3.

$$y=\frac{400}{3}$$

The system is now solved.

$$x=-10,y=\frac{400}{3}$$