Solve {l}{9m-28n=1}{72m-7n=1}

$$\left\{ \begin{array} { l } { 9 m - 28 n = 1 } \\ { 72 m - 7 n = 1 } \end{array} \right.$$

Solve for m, n

$$m=\frac{1}{93}\approx 0.010752688$$

$$n=-\frac{1}{31}\approx -0.032258065$$

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.

$$9m-28n=1,72m-7n=1$$

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

$$9m-28n=1$$

Add 28n to both sides of the equation.

$$9m=28n+1$$

Divide both sides by 9.

$$m=\frac{1}{9}\left(28n+1\right)$$

Multiply \frac{1}{9} times 28n+1.

$$m=\frac{28}{9}n+\frac{1}{9}$$

Substitute \frac{28n+1}{9} for m in the other equation, 72m-7n=1.

$$72\left(\frac{28}{9}n+\frac{1}{9}\right)-7n=1$$

Multiply 72 times \frac{28n+1}{9}.

$$224n+8-7n=1$$

Add 224n to -7n.

$$217n+8=1$$

Subtract 8 from both sides of the equation.

$$217n=-7$$

Divide both sides by 217.

$$n=-\frac{1}{31}$$

Substitute -\frac{1}{31} for n in m=\frac{28}{9}n+\frac{1}{9}. Because the resulting equation contains only one variable, you can solve for m directly.

$$m=\frac{28}{9}\left(-\frac{1}{31}\right)+\frac{1}{9}$$

Multiply \frac{28}{9} times -\frac{1}{31} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$m=-\frac{28}{279}+\frac{1}{9}$$

Add \frac{1}{9} to -\frac{28}{279} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$m=\frac{1}{93}$$

The system is now solved.

$$m=\frac{1}{93},n=-\frac{1}{31}$$

Steps Using Matrices

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

$$9m-28n=1,72m-7n=1$$

Write the equations in matrix form.

$$\left(\begin{matrix}9&-28\\72&-7\end{matrix}\right)\left(\begin{matrix}m\\n\end{matrix}\right)=\left(\begin{matrix}1\\1\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}9&-28\\72&-7\end{matrix}\right).

$$inverse(\left(\begin{matrix}9&-28\\72&-7\end{matrix}\right))\left(\begin{matrix}9&-28\\72&-7\end{matrix}\right)\left(\begin{matrix}m\\n\end{matrix}\right)=inverse(\left(\begin{matrix}9&-28\\72&-7\end{matrix}\right))\left(\begin{matrix}1\\1\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}m\\n\end{matrix}\right)=inverse(\left(\begin{matrix}9&-28\\72&-7\end{matrix}\right))\left(\begin{matrix}1\\1\end{matrix}\right)$$

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

$$\left(\begin{matrix}m\\n\end{matrix}\right)=inverse(\left(\begin{matrix}9&-28\\72&-7\end{matrix}\right))\left(\begin{matrix}1\\1\end{matrix}\right)$$

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.

$$\left(\begin{matrix}m\\n\end{matrix}\right)=\left(\begin{matrix}-\frac{7}{9\left(-7\right)-\left(-28\times 72\right)}&-\frac{-28}{9\left(-7\right)-\left(-28\times 72\right)}\\-\frac{72}{9\left(-7\right)-\left(-28\times 72\right)}&\frac{9}{9\left(-7\right)-\left(-28\times 72\right)}\end{matrix}\right)\left(\begin{matrix}1\\1\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}m\\n\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{279}&\frac{4}{279}\\-\frac{8}{217}&\frac{1}{217}\end{matrix}\right)\left(\begin{matrix}1\\1\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}m\\n\end{matrix}\right)=\left(\begin{matrix}\frac{-1+4}{279}\\\frac{-8+1}{217}\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}m\\n\end{matrix}\right)=\left(\begin{matrix}\frac{1}{93}\\-\frac{1}{31}\end{matrix}\right)$$

Extract the matrix elements m and n.

$$m=\frac{1}{93},n=-\frac{1}{31}$$

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.

$$9m-28n=1,72m-7n=1$$

To make 9m and 72m equal, multiply all terms on each side of the first equation by 72 and all terms on each side of the second by 9.

$$72\times 9m+72\left(-28\right)n=72,9\times 72m+9\left(-7\right)n=9$$

Simplify.

$$648m-2016n=72,648m-63n=9$$

Subtract 648m-63n=9 from 648m-2016n=72 by subtracting like terms on each side of the equal sign.

$$648m-648m-2016n+63n=72-9$$

Add 648m to -648m. Terms 648m and -648m cancel out, leaving an equation with only one variable that can be solved.

$$-2016n+63n=72-9$$

Add -2016n to 63n.

$$-1953n=72-9$$

Add 72 to -9.

$$-1953n=63$$

Divide both sides by -1953.

$$n=-\frac{1}{31}$$

Substitute -\frac{1}{31} for n in 72m-7n=1. Because the resulting equation contains only one variable, you can solve for m directly.

$$72m-7\left(-\frac{1}{31}\right)=1$$

Multiply -7 times -\frac{1}{31}.

$$72m+\frac{7}{31}=1$$

Subtract \frac{7}{31} from both sides of the equation.

$$72m=\frac{24}{31}$$

Divide both sides by 72.

$$m=\frac{1}{93}$$

The system is now solved.

$$m=\frac{1}{93},n=-\frac{1}{31}$$