Solve {l}{6x+7y=1320}{9x+4y=1200}

$$\left. \begin{array} { l } { 6 x + 7 y = 1320 } \\ { 9 x + 4 y = 1200 } \end{array} \right.$$

Solve for x, y

$$x=80$$

$$y=120$$

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.

$$6x+7y=1320,9x+4y=1200$$

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

$$6x+7y=1320$$

Subtract 7y from both sides of the equation.

$$6x=-7y+1320$$

Divide both sides by 6.

$$x=\frac{1}{6}\left(-7y+1320\right)$$

Multiply \frac{1}{6} times -7y+1320.

$$x=-\frac{7}{6}y+220$$

Substitute -\frac{7y}{6}+220 for x in the other equation, 9x+4y=1200.

$$9\left(-\frac{7}{6}y+220\right)+4y=1200$$

Multiply 9 times -\frac{7y}{6}+220.

$$-\frac{21}{2}y+1980+4y=1200$$

Add -\frac{21y}{2} to 4y.

$$-\frac{13}{2}y+1980=1200$$

Subtract 1980 from both sides of the equation.

$$-\frac{13}{2}y=-780$$

Divide both sides of the equation by -\frac{13}{2}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=120$$

Substitute 120 for y in x=-\frac{7}{6}y+220. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=-\frac{7}{6}\times 120+220$$

Multiply -\frac{7}{6} times 120.

$$x=-140+220$$

Add 220 to -140.

$$x=80$$

The system is now solved.

$$x=80,y=120$$

Steps Using Matrices

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

$$6x+7y=1320,9x+4y=1200$$

Write the equations in matrix form.

$$\left(\begin{matrix}6&7\\9&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1320\\1200\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}6&7\\9&4\end{matrix}\right).

$$inverse(\left(\begin{matrix}6&7\\9&4\end{matrix}\right))\left(\begin{matrix}6&7\\9&4\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&7\\9&4\end{matrix}\right))\left(\begin{matrix}1320\\1200\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&7\\9&4\end{matrix}\right))\left(\begin{matrix}1320\\1200\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}6&7\\9&4\end{matrix}\right))\left(\begin{matrix}1320\\1200\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{4}{6\times 4-7\times 9}&-\frac{7}{6\times 4-7\times 9}\\-\frac{9}{6\times 4-7\times 9}&\frac{6}{6\times 4-7\times 9}\end{matrix}\right)\left(\begin{matrix}1320\\1200\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{4}{39}&\frac{7}{39}\\\frac{3}{13}&-\frac{2}{13}\end{matrix}\right)\left(\begin{matrix}1320\\1200\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{4}{39}\times 1320+\frac{7}{39}\times 1200\\\frac{3}{13}\times 1320-\frac{2}{13}\times 1200\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}80\\120\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=80,y=120$$

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.

$$6x+7y=1320,9x+4y=1200$$

To make 6x and 9x equal, multiply all terms on each side of the first equation by 9 and all terms on each side of the second by 6.

$$9\times 6x+9\times 7y=9\times 1320,6\times 9x+6\times 4y=6\times 1200$$

Simplify.

$$54x+63y=11880,54x+24y=7200$$

Subtract 54x+24y=7200 from 54x+63y=11880 by subtracting like terms on each side of the equal sign.

$$54x-54x+63y-24y=11880-7200$$

Add 54x to -54x. Terms 54x and -54x cancel out, leaving an equation with only one variable that can be solved.

$$63y-24y=11880-7200$$

Add 63y to -24y.

$$39y=11880-7200$$

Add 11880 to -7200.

$$39y=4680$$

Divide both sides by 39.

$$y=120$$

Substitute 120 for y in 9x+4y=1200. Because the resulting equation contains only one variable, you can solve for x directly.

$$9x+4\times 120=1200$$

Multiply 4 times 120.

$$9x+480=1200$$

Subtract 480 from both sides of the equation.

$$9x=720$$

Divide both sides by 9.

$$x=80$$

The system is now solved.

$$x=80,y=120$$