Solve {l}{30x+20y=240}{30y-40x=20}

$$\left\{ \begin{array} { l } { 30 x + 20 y = 240 } \\ { 30 y - 40 x = 20 } \end{array} \right.$$

Solve for x, y

$$x=4$$

$$y=6$$

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.

$$30x+20y=240,-40x+30y=20$$

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

$$30x+20y=240$$

Subtract 20y from both sides of the equation.

$$30x=-20y+240$$

Divide both sides by 30.

$$x=\frac{1}{30}\left(-20y+240\right)$$

Multiply \frac{1}{30} times -20y+240.

$$x=-\frac{2}{3}y+8$$

Substitute -\frac{2y}{3}+8 for x in the other equation, -40x+30y=20.

$$-40\left(-\frac{2}{3}y+8\right)+30y=20$$

Multiply -40 times -\frac{2y}{3}+8.

$$\frac{80}{3}y-320+30y=20$$

Add \frac{80y}{3} to 30y.

$$\frac{170}{3}y-320=20$$

Add 320 to both sides of the equation.

$$\frac{170}{3}y=340$$

Divide both sides of the equation by \frac{170}{3}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=6$$

Substitute 6 for y in x=-\frac{2}{3}y+8. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-\frac{2}{3}\times 6+8$$

Multiply -\frac{2}{3} times 6.

$$x=-4+8$$

Add 8 to -4.

$$x=4$$

The system is now solved.

$$x=4,y=6$$

Steps Using Matrices

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

$$30x+20y=240,-40x+30y=20$$

Write the equations in matrix form.

$$\left(\begin{matrix}30&20\\-40&30\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}240\\20\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}30&20\\-40&30\end{matrix}\right).

$$inverse(\left(\begin{matrix}30&20\\-40&30\end{matrix}\right))\left(\begin{matrix}30&20\\-40&30\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}30&20\\-40&30\end{matrix}\right))\left(\begin{matrix}240\\20\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}30&20\\-40&30\end{matrix}\right))\left(\begin{matrix}240\\20\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}30&20\\-40&30\end{matrix}\right))\left(\begin{matrix}240\\20\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{30}{30\times 30-20\left(-40\right)}&-\frac{20}{30\times 30-20\left(-40\right)}\\-\frac{-40}{30\times 30-20\left(-40\right)}&\frac{30}{30\times 30-20\left(-40\right)}\end{matrix}\right)\left(\begin{matrix}240\\20\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{170}&-\frac{1}{85}\\\frac{2}{85}&\frac{3}{170}\end{matrix}\right)\left(\begin{matrix}240\\20\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{170}\times 240-\frac{1}{85}\times 20\\\frac{2}{85}\times 240+\frac{3}{170}\times 20\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}4\\6\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=4,y=6$$

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.

$$30x+20y=240,-40x+30y=20$$

To make 30x and -40x equal, multiply all terms on each side of the first equation by -40 and all terms on each side of the second by 30.

$$-40\times 30x-40\times 20y=-40\times 240,30\left(-40\right)x+30\times 30y=30\times 20$$

Simplify.

$$-1200x-800y=-9600,-1200x+900y=600$$

Subtract -1200x+900y=600 from -1200x-800y=-9600 by subtracting like terms on each side of the equal sign.

$$-1200x+1200x-800y-900y=-9600-600$$

Add -1200x to 1200x. Terms -1200x and 1200x cancel out, leaving an equation with only one variable that can be solved.

$$-800y-900y=-9600-600$$

Add -800y to -900y.

$$-1700y=-9600-600$$

Add -9600 to -600.

$$-1700y=-10200$$

Divide both sides by -1700.

$$y=6$$

Substitute 6 for y in -40x+30y=20. Because the resulting equation contains only one variable, you can solve for x directly.

$$-40x+30\times 6=20$$

Multiply 30 times 6.

$$-40x+180=20$$

Subtract 180 from both sides of the equation.

$$-40x=-160$$

Divide both sides by -40.

$$x=4$$

The system is now solved.

$$x=4,y=6$$