Solve {c}{2.7232*100+n*2.7170*104+m*2.7125*94=0}{10.2056*100+n*101867*104+m*100737*94=0}

$$\left\{ \begin{array} { c } { 2.7232 * 100 + n * 2.7170 * 104 + m * 2.7125 * 94 = 0 } \\ { 10.2056 * 100 + n * 101867 * 104 + m * 100737 * 94 = 0 } \end{array} \right.$$

Solve for n, m

$$n = \frac{3918561653}{38804012} = 100\frac{38160453}{38804012} \approx 100.983415143$$

$$m = -\frac{693441214462}{6137749975} = -112\frac{6013217262}{6137749975} \approx -112.979710364$$

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.

$$282.568n+254.975m+272.32=0,10594168n+9469278m+1020.56=0$$

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

$$282.568n+254.975m+272.32=0$$

Subtract 272.32 from both sides of the equation.

$$282.568n+254.975m=-272.32$$

Subtract \frac{10199m}{40} from both sides of the equation.

$$282.568n=-254.975m-272.32$$

Divide both sides of the equation by 282.568, which is the same as multiplying both sides by the reciprocal of the fraction.

$$n=\frac{125}{35321}\left(-254.975m-272.32\right)$$

Multiply \frac{125}{35321} times -\frac{10199m}{40}-272.32.

$$n=-\frac{254975}{282568}m-\frac{34040}{35321}$$

Substitute -\frac{254975m}{282568}-\frac{34040}{35321} for n in the other equation, 10594168n+9469278m+1020.56=0.

$$10594168\left(-\frac{254975}{282568}m-\frac{34040}{35321}\right)+9469278m+1020.56=0$$

Multiply 10594168 times -\frac{254975m}{282568}-\frac{34040}{35321}.

$$-\frac{25973538325}{2717}m-\frac{27740421440}{2717}+9469278m+1020.56=0$$

Add -\frac{25973538325m}{2717} to 9469278m.

$$-\frac{245509999}{2717}m-\frac{27740421440}{2717}+1020.56=0$$

Add -\frac{27740421440}{2717} to 1020.56 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$-\frac{245509999}{2717}m-\frac{693441214462}{67925}=0$$

Add \frac{693441214462}{67925} to both sides of the equation.

$$-\frac{245509999}{2717}m=\frac{693441214462}{67925}$$

Divide both sides of the equation by -\frac{245509999}{2717}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$m=-\frac{693441214462}{6137749975}$$

Substitute -\frac{693441214462}{6137749975} for m in n=-\frac{254975}{282568}m-\frac{34040}{35321}. Because the resulting equation contains only one variable, you can solve for n directly.

$$n=-\frac{254975}{282568}\left(-\frac{693441214462}{6137749975}\right)-\frac{34040}{35321}$$

Multiply -\frac{254975}{282568} times -\frac{693441214462}{6137749975} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$n=\frac{10748338824161}{105430500604}-\frac{34040}{35321}$$

Add -\frac{34040}{35321} to \frac{10748338824161}{105430500604} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$n=\frac{3918561653}{38804012}$$

The system is now solved.

$$n=\frac{3918561653}{38804012},m=-\frac{693441214462}{6137749975}$$

Steps Using Matrices

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

$$282.568n+254.975m+272.32=0,10594168n+9469278m+1020.56=0$$

Write the equations in matrix form.

$$\left(\begin{matrix}282.568&254.975\\10594168&9469278\end{matrix}\right)\left(\begin{matrix}n\\m\end{matrix}\right)=\left(\begin{matrix}-272.32\\-1020.56\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}282.568&254.975\\10594168&9469278\end{matrix}\right).

$$inverse(\left(\begin{matrix}282.568&254.975\\10594168&9469278\end{matrix}\right))\left(\begin{matrix}282.568&254.975\\10594168&9469278\end{matrix}\right)\left(\begin{matrix}n\\m\end{matrix}\right)=inverse(\left(\begin{matrix}282.568&254.975\\10594168&9469278\end{matrix}\right))\left(\begin{matrix}-272.32\\-1020.56\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}n\\m\end{matrix}\right)=inverse(\left(\begin{matrix}282.568&254.975\\10594168&9469278\end{matrix}\right))\left(\begin{matrix}-272.32\\-1020.56\end{matrix}\right)$$

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

$$\left(\begin{matrix}n\\m\end{matrix}\right)=inverse(\left(\begin{matrix}282.568&254.975\\10594168&9469278\end{matrix}\right))\left(\begin{matrix}-272.32\\-1020.56\end{matrix}\right)$$

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.

$$\left(\begin{matrix}n\\m\end{matrix}\right)=\left(\begin{matrix}\frac{9469278}{282.568\times 9469278-254.975\times 10594168}&-\frac{254.975}{282.568\times 9469278-254.975\times 10594168}\\-\frac{10594168}{282.568\times 9469278-254.975\times 10594168}&\frac{282.568}{282.568\times 9469278-254.975\times 10594168}\end{matrix}\right)\left(\begin{matrix}-272.32\\-1020.56\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}n\\m\end{matrix}\right)=\left(\begin{matrix}-\frac{276750}{746231}&\frac{775}{77608024}\\\frac{101867000}{245509999}&-\frac{2717}{245509999}\end{matrix}\right)\left(\begin{matrix}-272.32\\-1020.56\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}n\\m\end{matrix}\right)=\left(\begin{matrix}-\frac{276750}{746231}\left(-272.32\right)+\frac{775}{77608024}\left(-1020.56\right)\\\frac{101867000}{245509999}\left(-272.32\right)-\frac{2717}{245509999}\left(-1020.56\right)\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}n\\m\end{matrix}\right)=\left(\begin{matrix}\frac{3918561653}{38804012}\\-\frac{693441214462}{6137749975}\end{matrix}\right)$$

Extract the matrix elements n and m.

$$n=\frac{3918561653}{38804012},m=-\frac{693441214462}{6137749975}$$

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.

$$282.568n+254.975m+272.32=0,10594168n+9469278m+1020.56=0$$

To make \frac{35321n}{125} and 10594168n equal, multiply all terms on each side of the first equation by 10594168 and all terms on each side of the second by 282.568.

$$10594168\times 282.568n+10594168\times 254.975m+10594168\times 272.32=0,282.568\times 10594168n+282.568\times 9469278m+282.568\times 1020.56=0$$

Simplify.

$$2993572863.424n+2701247985.8m+2885003829.76=0,2993572863.424n+2675714945.904m+288377.59808=0$$

Subtract 2993572863.424n+2675714945.904m+288377.59808=0 from 2993572863.424n+2701247985.8m+2885003829.76=0 by subtracting like terms on each side of the equal sign.

$$2993572863.424n-2993572863.424n+2701247985.8m-2675714945.904m+2885003829.76-288377.59808=0$$

Add \frac{374196607928n}{125} to -\frac{374196607928n}{125}. Terms \frac{374196607928n}{125} and -\frac{374196607928n}{125} cancel out, leaving an equation with only one variable that can be solved.

$$2701247985.8m-2675714945.904m+2885003829.76-288377.59808=0$$

Add \frac{13506239929m}{5} to -\frac{334464368238m}{125}.

$$25533039.896m+2885003829.76-288377.59808=0$$

Add 2885003829.76 to -288377.59808 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$25533039.896m+2884715452.16192=0$$

Subtract 2884715452.16192 from both sides of the equation.

$$25533039.896m=-2884715452.16192$$

Divide both sides of the equation by 25533039.896, which is the same as multiplying both sides by the reciprocal of the fraction.

$$m=-\frac{693441214462}{6137749975}$$

Substitute -\frac{693441214462}{6137749975} for m in 10594168n+9469278m+1020.56=0. Because the resulting equation contains only one variable, you can solve for n directly.

$$10594168n+9469278\left(-\frac{693441214462}{6137749975}\right)+1020.56=0$$

Multiply 9469278 times -\frac{693441214462}{6137749975}.

$$10594168n-\frac{19958625034645284}{18655775}+1020.56=0$$

Add -\frac{19958625034645284}{18655775} to 1020.56 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$10594168n-\frac{798344239812302}{746231}=0$$

Add \frac{798344239812302}{746231} to both sides of the equation.

$$10594168n=\frac{798344239812302}{746231}$$

Divide both sides by 10594168.

$$n=\frac{3918561653}{38804012}$$

The system is now solved.

$$n=\frac{3918561653}{38804012},m=-\frac{693441214462}{6137749975}$$