Solve {l}{8x+10y=14400}{12x+14y=12000}

$$\left. \begin{array} { l } { 8 x + 10 y = 14400 } \\ { 12 x + 14 y = 12000 } \end{array} \right.$$

Solve for x, y

$$x=-10200$$

$$y=9600$$

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.

$$8x+10y=14400,12x+14y=12000$$

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

$$8x+10y=14400$$

Subtract 10y from both sides of the equation.

$$8x=-10y+14400$$

Divide both sides by 8.

$$x=\frac{1}{8}\left(-10y+14400\right)$$

Multiply \frac{1}{8} times -10y+14400.

$$x=-\frac{5}{4}y+1800$$

Substitute -\frac{5y}{4}+1800 for x in the other equation, 12x+14y=12000.

$$12\left(-\frac{5}{4}y+1800\right)+14y=12000$$

Multiply 12 times -\frac{5y}{4}+1800.

$$-15y+21600+14y=12000$$

Add -15y to 14y.

$$-y+21600=12000$$

Subtract 21600 from both sides of the equation.

$$-y=-9600$$

Divide both sides by -1.

$$y=9600$$

Substitute 9600 for y in x=-\frac{5}{4}y+1800. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-\frac{5}{4}\times 9600+1800$$

Multiply -\frac{5}{4} times 9600.

$$x=-12000+1800$$

Add 1800 to -12000.

$$x=-10200$$

The system is now solved.

$$x=-10200,y=9600$$

Steps Using Matrices

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

$$8x+10y=14400,12x+14y=12000$$

Write the equations in matrix form.

$$\left(\begin{matrix}8&10\\12&14\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}14400\\12000\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}8&10\\12&14\end{matrix}\right).

$$inverse(\left(\begin{matrix}8&10\\12&14\end{matrix}\right))\left(\begin{matrix}8&10\\12&14\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&10\\12&14\end{matrix}\right))\left(\begin{matrix}14400\\12000\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&10\\12&14\end{matrix}\right))\left(\begin{matrix}14400\\12000\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}8&10\\12&14\end{matrix}\right))\left(\begin{matrix}14400\\12000\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{14}{8\times 14-10\times 12}&-\frac{10}{8\times 14-10\times 12}\\-\frac{12}{8\times 14-10\times 12}&\frac{8}{8\times 14-10\times 12}\end{matrix}\right)\left(\begin{matrix}14400\\12000\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{7}{4}&\frac{5}{4}\\\frac{3}{2}&-1\end{matrix}\right)\left(\begin{matrix}14400\\12000\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{7}{4}\times 14400+\frac{5}{4}\times 12000\\\frac{3}{2}\times 14400-12000\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-10200\\9600\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=-10200,y=9600$$

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.

$$8x+10y=14400,12x+14y=12000$$

To make 8x and 12x equal, multiply all terms on each side of the first equation by 12 and all terms on each side of the second by 8.

$$12\times 8x+12\times 10y=12\times 14400,8\times 12x+8\times 14y=8\times 12000$$

Simplify.

$$96x+120y=172800,96x+112y=96000$$

Subtract 96x+112y=96000 from 96x+120y=172800 by subtracting like terms on each side of the equal sign.

$$96x-96x+120y-112y=172800-96000$$

Add 96x to -96x. Terms 96x and -96x cancel out, leaving an equation with only one variable that can be solved.

$$120y-112y=172800-96000$$

Add 120y to -112y.

$$8y=172800-96000$$

Add 172800 to -96000.

$$8y=76800$$

Divide both sides by 8.

$$y=9600$$

Substitute 9600 for y in 12x+14y=12000. Because the resulting equation contains only one variable, you can solve for x directly.

$$12x+14\times 9600=12000$$

Multiply 14 times 9600.

$$12x+134400=12000$$

Subtract 134400 from both sides of the equation.

$$12x=-122400$$

Divide both sides by 12.

$$x=-10200$$

The system is now solved.

$$x=-10200,y=9600$$