Solve {c}{12x+4y=14,280}{8x+6y=9,040}

$$\left. \begin{array} { c } { 12 x + 4 y = 14,280 } \\ { 8 x + 6 y = 9,040 } \end{array} \right.$$

Solve for x, y

$$x=1238$$

$$y=-144$$

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.

$$12x+4y=14280,8x+6y=9040$$

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

$$12x+4y=14280$$

Subtract 4y from both sides of the equation.

$$12x=-4y+14280$$

Divide both sides by 12.

$$x=\frac{1}{12}\left(-4y+14280\right)$$

Multiply \frac{1}{12} times -4y+14280.

$$x=-\frac{1}{3}y+1190$$

Substitute -\frac{y}{3}+1190 for x in the other equation, 8x+6y=9040.

$$8\left(-\frac{1}{3}y+1190\right)+6y=9040$$

Multiply 8 times -\frac{y}{3}+1190.

$$-\frac{8}{3}y+9520+6y=9040$$

Add -\frac{8y}{3} to 6y.

$$\frac{10}{3}y+9520=9040$$

Subtract 9520 from both sides of the equation.

$$\frac{10}{3}y=-480$$

Divide both sides of the equation by \frac{10}{3}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=-144$$

Substitute -144 for y in x=-\frac{1}{3}y+1190. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-\frac{1}{3}\left(-144\right)+1190$$

Multiply -\frac{1}{3} times -144.

$$x=48+1190$$

Add 1190 to 48.

$$x=1238$$

The system is now solved.

$$x=1238,y=-144$$

Steps Using Matrices

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

$$12x+4y=14280,8x+6y=9040$$

Write the equations in matrix form.

$$\left(\begin{matrix}12&4\\8&6\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}14280\\9040\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}12&4\\8&6\end{matrix}\right).

$$inverse(\left(\begin{matrix}12&4\\8&6\end{matrix}\right))\left(\begin{matrix}12&4\\8&6\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}12&4\\8&6\end{matrix}\right))\left(\begin{matrix}14280\\9040\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}12&4\\8&6\end{matrix}\right))\left(\begin{matrix}14280\\9040\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}12&4\\8&6\end{matrix}\right))\left(\begin{matrix}14280\\9040\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{6}{12\times 6-4\times 8}&-\frac{4}{12\times 6-4\times 8}\\-\frac{8}{12\times 6-4\times 8}&\frac{12}{12\times 6-4\times 8}\end{matrix}\right)\left(\begin{matrix}14280\\9040\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{20}&-\frac{1}{10}\\-\frac{1}{5}&\frac{3}{10}\end{matrix}\right)\left(\begin{matrix}14280\\9040\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{20}\times 14280-\frac{1}{10}\times 9040\\-\frac{1}{5}\times 14280+\frac{3}{10}\times 9040\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1238\\-144\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=1238,y=-144$$

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.

$$12x+4y=14280,8x+6y=9040$$

To make 12x 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 12.

$$8\times 12x+8\times 4y=8\times 14280,12\times 8x+12\times 6y=12\times 9040$$

Simplify.

$$96x+32y=114240,96x+72y=108480$$

Subtract 96x+72y=108480 from 96x+32y=114240 by subtracting like terms on each side of the equal sign.

$$96x-96x+32y-72y=114240-108480$$

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

$$32y-72y=114240-108480$$

Add 32y to -72y.

$$-40y=114240-108480$$

Add 114240 to -108480.

$$-40y=5760$$

Divide both sides by -40.

$$y=-144$$

Substitute -144 for y in 8x+6y=9040. Because the resulting equation contains only one variable, you can solve for x directly.

$$8x+6\left(-144\right)=9040$$

Multiply 6 times -144.

$$8x-864=9040$$

Add 864 to both sides of the equation.

$$8x=9904$$

Divide both sides by 8.

$$x=1238$$

The system is now solved.

$$x=1238,y=-144$$