Solve {l}{6x+15y=360}{8x+10y=440}

$$\left\{ \begin{array} { l } { 6 x + 15 y = 360 } \\ { 8 x + 10 y = 440 } \end{array} \right.$$

Solve for x, y

$$x=50$$

$$y=4$$

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.

$$6x+15y=360,8x+10y=440$$

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

$$6x+15y=360$$

Subtract 15y from both sides of the equation.

$$6x=-15y+360$$

Divide both sides by 6.

$$x=\frac{1}{6}\left(-15y+360\right)$$

Multiply \frac{1}{6} times -15y+360.

$$x=-\frac{5}{2}y+60$$

Substitute -\frac{5y}{2}+60 for x in the other equation, 8x+10y=440.

$$8\left(-\frac{5}{2}y+60\right)+10y=440$$

Multiply 8 times -\frac{5y}{2}+60.

$$-20y+480+10y=440$$

Add -20y to 10y.

$$-10y+480=440$$

Subtract 480 from both sides of the equation.

$$-10y=-40$$

Divide both sides by -10.

$$y=4$$

Substitute 4 for y in x=-\frac{5}{2}y+60. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-\frac{5}{2}\times 4+60$$

Multiply -\frac{5}{2} times 4.

$$x=-10+60$$

Add 60 to -10.

$$x=50$$

The system is now solved.

$$x=50,y=4$$

Steps Using Matrices

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

$$6x+15y=360,8x+10y=440$$

Write the equations in matrix form.

$$\left(\begin{matrix}6&15\\8&10\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}360\\440\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}6&15\\8&10\end{matrix}\right).

$$inverse(\left(\begin{matrix}6&15\\8&10\end{matrix}\right))\left(\begin{matrix}6&15\\8&10\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&15\\8&10\end{matrix}\right))\left(\begin{matrix}360\\440\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&15\\8&10\end{matrix}\right))\left(\begin{matrix}360\\440\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&15\\8&10\end{matrix}\right))\left(\begin{matrix}360\\440\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{10}{6\times 10-15\times 8}&-\frac{15}{6\times 10-15\times 8}\\-\frac{8}{6\times 10-15\times 8}&\frac{6}{6\times 10-15\times 8}\end{matrix}\right)\left(\begin{matrix}360\\440\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{6}&\frac{1}{4}\\\frac{2}{15}&-\frac{1}{10}\end{matrix}\right)\left(\begin{matrix}360\\440\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{6}\times 360+\frac{1}{4}\times 440\\\frac{2}{15}\times 360-\frac{1}{10}\times 440\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}50\\4\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=50,y=4$$

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.

$$6x+15y=360,8x+10y=440$$

To make 6x and 8x equal, multiply all terms on each side of the first equation by 8 and all terms on each side of the second by 6.

$$8\times 6x+8\times 15y=8\times 360,6\times 8x+6\times 10y=6\times 440$$

Simplify.

$$48x+120y=2880,48x+60y=2640$$

Subtract 48x+60y=2640 from 48x+120y=2880 by subtracting like terms on each side of the equal sign.

$$48x-48x+120y-60y=2880-2640$$

Add 48x to -48x. Terms 48x and -48x cancel out, leaving an equation with only one variable that can be solved.

$$120y-60y=2880-2640$$

Add 120y to -60y.

$$60y=2880-2640$$

Add 2880 to -2640.

$$60y=240$$

Divide both sides by 60.

$$y=4$$

Substitute 4 for y in 8x+10y=440. Because the resulting equation contains only one variable, you can solve for x directly.

$$8x+10\times 4=440$$

Multiply 10 times 4.

$$8x+40=440$$

Subtract 40 from both sides of the equation.

$$8x=400$$

Divide both sides by 8.

$$x=50$$

The system is now solved.

$$x=50,y=4$$