Solve {l}{9x+8y=5280x?}{8x+12y=5280*9}

$$\left. \begin{array} { l } { 9 x + 8 y = 5280 x ? } \\ { 8 x + 12 y = 5280 \times 9 } \end{array} \right.$$

Solve for x, y

$$x = \frac{8640}{1439} = 6\frac{6}{1439} \approx 6.004169562$$

$$y = \frac{5692680}{1439} = 3955\frac{1435}{1439} \approx 3955.997220292$$

Steps Using Substitution

Consider the first equation. Subtract 5280x from both sides.

$$9x+8y-5280x=0$$

Combine 9x and -5280x to get -5271x.

$$-5271x+8y=0$$

Consider the second equation. Multiply 5280 and 9 to get 47520.

$$8x+12y=47520$$

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.

$$-5271x+8y=0,8x+12y=47520$$

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

$$-5271x+8y=0$$

Subtract 8y from both sides of the equation.

$$-5271x=-8y$$

Divide both sides by -5271.

$$x=-\frac{1}{5271}\left(-8\right)y$$

Multiply -\frac{1}{5271} times -8y.

$$x=\frac{8}{5271}y$$

Substitute \frac{8y}{5271} for x in the other equation, 8x+12y=47520.

$$8\times \frac{8}{5271}y+12y=47520$$

Multiply 8 times \frac{8y}{5271}.

$$\frac{64}{5271}y+12y=47520$$

Add \frac{64y}{5271} to 12y.

$$\frac{63316}{5271}y=47520$$

Divide both sides of the equation by \frac{63316}{5271}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$y=\frac{5692680}{1439}$$

Substitute \frac{5692680}{1439} for y in x=\frac{8}{5271}y. Because the resulting equation contains only one variable, you can solve for x directly.

$$x=\frac{8}{5271}\times \frac{5692680}{1439}$$

Multiply \frac{8}{5271} times \frac{5692680}{1439} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{8640}{1439}$$

The system is now solved.

$$x=\frac{8640}{1439},y=\frac{5692680}{1439}$$

Steps Using Matrices

Consider the first equation. Subtract 5280x from both sides.

$$9x+8y-5280x=0$$

Combine 9x and -5280x to get -5271x.

$$-5271x+8y=0$$

Consider the second equation. Multiply 5280 and 9 to get 47520.

$$8x+12y=47520$$

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

$$-5271x+8y=0,8x+12y=47520$$

Write the equations in matrix form.

$$\left(\begin{matrix}-5271&8\\8&12\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}0\\47520\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}-5271&8\\8&12\end{matrix}\right).

$$inverse(\left(\begin{matrix}-5271&8\\8&12\end{matrix}\right))\left(\begin{matrix}-5271&8\\8&12\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-5271&8\\8&12\end{matrix}\right))\left(\begin{matrix}0\\47520\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-5271&8\\8&12\end{matrix}\right))\left(\begin{matrix}0\\47520\end{matrix}\right)$$

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

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-5271&8\\8&12\end{matrix}\right))\left(\begin{matrix}0\\47520\end{matrix}\right)$$

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.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{12}{-5271\times 12-8\times 8}&-\frac{8}{-5271\times 12-8\times 8}\\-\frac{8}{-5271\times 12-8\times 8}&-\frac{5271}{-5271\times 12-8\times 8}\end{matrix}\right)\left(\begin{matrix}0\\47520\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}-\frac{3}{15829}&\frac{2}{15829}\\\frac{2}{15829}&\frac{5271}{63316}\end{matrix}\right)\left(\begin{matrix}0\\47520\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{2}{15829}\times 47520\\\frac{5271}{63316}\times 47520\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{8640}{1439}\\\frac{5692680}{1439}\end{matrix}\right)$$

Extract the matrix elements x and y.

$$x=\frac{8640}{1439},y=\frac{5692680}{1439}$$

Steps Using Elimination

Consider the first equation. Subtract 5280x from both sides.

$$9x+8y-5280x=0$$

Combine 9x and -5280x to get -5271x.

$$-5271x+8y=0$$

Consider the second equation. Multiply 5280 and 9 to get 47520.

$$8x+12y=47520$$

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.

$$-5271x+8y=0,8x+12y=47520$$

To make -5271x 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 -5271.

$$8\left(-5271\right)x+8\times 8y=0,-5271\times 8x-5271\times 12y=-5271\times 47520$$

Simplify.

$$-42168x+64y=0,-42168x-63252y=-250477920$$

Subtract -42168x-63252y=-250477920 from -42168x+64y=0 by subtracting like terms on each side of the equal sign.

$$-42168x+42168x+64y+63252y=250477920$$

Add -42168x to 42168x. Terms -42168x and 42168x cancel out, leaving an equation with only one variable that can be solved.

$$64y+63252y=250477920$$

Add 64y to 63252y.

$$63316y=250477920$$

Divide both sides by 63316.

$$y=\frac{5692680}{1439}$$

Substitute \frac{5692680}{1439} for y in 8x+12y=47520. Because the resulting equation contains only one variable, you can solve for x directly.

$$8x+12\times \frac{5692680}{1439}=47520$$

Multiply 12 times \frac{5692680}{1439}.

$$8x+\frac{68312160}{1439}=47520$$

Subtract \frac{68312160}{1439} from both sides of the equation.

$$8x=\frac{69120}{1439}$$

Divide both sides by 8.

$$x=\frac{8640}{1439}$$

The system is now solved.

$$x=\frac{8640}{1439},y=\frac{5692680}{1439}$$