Solve {l}{10x+20y=240}{10y+20x=240}

$$\left. \begin{array} { l } { 10 x + 20 y = 240 } \\ { 10 y + 20 x = 240 } \end{array} \right.$$

Solve for x, y

$$x=8$$

$$y=8$$

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.

$$10x+20y=240,20x+10y=240$$

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

$$10x+20y=240$$

Subtract 20y from both sides of the equation.

$$10x=-20y+240$$

Divide both sides by 10.

$$x=\frac{1}{10}\left(-20y+240\right)$$

Multiply \frac{1}{10} times -20y+240.

$$x=-2y+24$$

Substitute -2y+24 for x in the other equation, 20x+10y=240.

$$20\left(-2y+24\right)+10y=240$$

Multiply 20 times -2y+24.

$$-40y+480+10y=240$$

Add -40y to 10y.

$$-30y+480=240$$

Subtract 480 from both sides of the equation.

$$-30y=-240$$

Divide both sides by -30.

$$y=8$$

Substitute 8 for y in x=-2y+24. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-2\times 8+24$$

Multiply -2 times 8.

$$x=-16+24$$

Add 24 to -16.

$$x=8$$

The system is now solved.

$$x=8,y=8$$

Steps Using Matrices

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

$$10x+20y=240,20x+10y=240$$

Write the equations in matrix form.

$$\left(\begin{matrix}10&20\\20&10\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}240\\240\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}10&20\\20&10\end{matrix}\right).

$$inverse(\left(\begin{matrix}10&20\\20&10\end{matrix}\right))\left(\begin{matrix}10&20\\20&10\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}10&20\\20&10\end{matrix}\right))\left(\begin{matrix}240\\240\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}10&20\\20&10\end{matrix}\right))\left(\begin{matrix}240\\240\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}10&20\\20&10\end{matrix}\right))\left(\begin{matrix}240\\240\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{10}{10\times 10-20\times 20}&-\frac{20}{10\times 10-20\times 20}\\-\frac{20}{10\times 10-20\times 20}&\frac{10}{10\times 10-20\times 20}\end{matrix}\right)\left(\begin{matrix}240\\240\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{30}&\frac{1}{15}\\\frac{1}{15}&-\frac{1}{30}\end{matrix}\right)\left(\begin{matrix}240\\240\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{30}\times 240+\frac{1}{15}\times 240\\\frac{1}{15}\times 240-\frac{1}{30}\times 240\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}8\\8\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=8,y=8$$

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.

$$10x+20y=240,20x+10y=240$$

To make 10x and 20x equal, multiply all terms on each side of the first equation by 20 and all terms on each side of the second by 10.

$$20\times 10x+20\times 20y=20\times 240,10\times 20x+10\times 10y=10\times 240$$

Simplify.

$$200x+400y=4800,200x+100y=2400$$

Subtract 200x+100y=2400 from 200x+400y=4800 by subtracting like terms on each side of the equal sign.

$$200x-200x+400y-100y=4800-2400$$

Add 200x to -200x. Terms 200x and -200x cancel out, leaving an equation with only one variable that can be solved.

$$400y-100y=4800-2400$$

Add 400y to -100y.

$$300y=4800-2400$$

Add 4800 to -2400.

$$300y=2400$$

Divide both sides by 300.

$$y=8$$

Substitute 8 for y in 20x+10y=240. Because the resulting equation contains only one variable, you can solve for x directly.

$$20x+10\times 8=240$$

Multiply 10 times 8.

$$20x+80=240$$

Subtract 80 from both sides of the equation.

$$20x=160$$

Divide both sides by 20.

$$x=8$$

The system is now solved.

$$x=8,y=8$$