Solve {l}{9c+4100m=3264}{4100c+1994800m=1537280}

$$\left. \begin{array} { l } { 9 c + 4100 m = 3264 } \\ { 4100 c + 1994800 m = 1537280 } \end{array} \right.$$

Solve for c, m

$$c = \frac{260224}{1429} = 182\frac{146}{1429} \approx 182.102169349$$

$$m=\frac{2832}{7145}\approx 0.396361092$$

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.

$$9c+4100m=3264,4100c+1994800m=1537280$$

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

$$9c+4100m=3264$$

Subtract 4100m from both sides of the equation.

$$9c=-4100m+3264$$

Divide both sides by 9.

$$c=\frac{1}{9}\left(-4100m+3264\right)$$

Multiply \frac{1}{9} times -4100m+3264.

$$c=-\frac{4100}{9}m+\frac{1088}{3}$$

Substitute -\frac{4100m}{9}+\frac{1088}{3} for c in the other equation, 4100c+1994800m=1537280.

$$4100\left(-\frac{4100}{9}m+\frac{1088}{3}\right)+1994800m=1537280$$

Multiply 4100 times -\frac{4100m}{9}+\frac{1088}{3}.

$$-\frac{16810000}{9}m+\frac{4460800}{3}+1994800m=1537280$$

Add -\frac{16810000m}{9} to 1994800m.

$$\frac{1143200}{9}m+\frac{4460800}{3}=1537280$$

Subtract \frac{4460800}{3} from both sides of the equation.

$$\frac{1143200}{9}m=\frac{151040}{3}$$

Divide both sides of the equation by \frac{1143200}{9}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$m=\frac{2832}{7145}$$

Substitute \frac{2832}{7145} for m in c=-\frac{4100}{9}m+\frac{1088}{3}. Because the resulting equation contains only one variable, you can solve for c directly.

$$c=-\frac{4100}{9}\times \frac{2832}{7145}+\frac{1088}{3}$$

Multiply -\frac{4100}{9} times \frac{2832}{7145} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$c=-\frac{774080}{4287}+\frac{1088}{3}$$

Add \frac{1088}{3} to -\frac{774080}{4287} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$c=\frac{260224}{1429}$$

The system is now solved.

$$c=\frac{260224}{1429},m=\frac{2832}{7145}$$

Steps Using Matrices

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

$$9c+4100m=3264,4100c+1994800m=1537280$$

Write the equations in matrix form.

$$\left(\begin{matrix}9&4100\\4100&1994800\end{matrix}\right)\left(\begin{matrix}c\\m\end{matrix}\right)=\left(\begin{matrix}3264\\1537280\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}9&4100\\4100&1994800\end{matrix}\right).

$$inverse(\left(\begin{matrix}9&4100\\4100&1994800\end{matrix}\right))\left(\begin{matrix}9&4100\\4100&1994800\end{matrix}\right)\left(\begin{matrix}c\\m\end{matrix}\right)=inverse(\left(\begin{matrix}9&4100\\4100&1994800\end{matrix}\right))\left(\begin{matrix}3264\\1537280\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}c\\m\end{matrix}\right)=inverse(\left(\begin{matrix}9&4100\\4100&1994800\end{matrix}\right))\left(\begin{matrix}3264\\1537280\end{matrix}\right)$$

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

$$\left(\begin{matrix}c\\m\end{matrix}\right)=inverse(\left(\begin{matrix}9&4100\\4100&1994800\end{matrix}\right))\left(\begin{matrix}3264\\1537280\end{matrix}\right)$$

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.

$$\left(\begin{matrix}c\\m\end{matrix}\right)=\left(\begin{matrix}\frac{1994800}{9\times 1994800-4100\times 4100}&-\frac{4100}{9\times 1994800-4100\times 4100}\\-\frac{4100}{9\times 1994800-4100\times 4100}&\frac{9}{9\times 1994800-4100\times 4100}\end{matrix}\right)\left(\begin{matrix}3264\\1537280\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}c\\m\end{matrix}\right)=\left(\begin{matrix}\frac{4987}{2858}&-\frac{41}{11432}\\-\frac{41}{11432}&\frac{9}{1143200}\end{matrix}\right)\left(\begin{matrix}3264\\1537280\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}c\\m\end{matrix}\right)=\left(\begin{matrix}\frac{4987}{2858}\times 3264-\frac{41}{11432}\times 1537280\\-\frac{41}{11432}\times 3264+\frac{9}{1143200}\times 1537280\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}c\\m\end{matrix}\right)=\left(\begin{matrix}\frac{260224}{1429}\\\frac{2832}{7145}\end{matrix}\right)$$

Extract the matrix elements c and m.

$$c=\frac{260224}{1429},m=\frac{2832}{7145}$$

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.

$$9c+4100m=3264,4100c+1994800m=1537280$$

To make 9c and 4100c equal, multiply all terms on each side of the first equation by 4100 and all terms on each side of the second by 9.

$$4100\times 9c+4100\times 4100m=4100\times 3264,9\times 4100c+9\times 1994800m=9\times 1537280$$

Simplify.

$$36900c+16810000m=13382400,36900c+17953200m=13835520$$

Subtract 36900c+17953200m=13835520 from 36900c+16810000m=13382400 by subtracting like terms on each side of the equal sign.

$$36900c-36900c+16810000m-17953200m=13382400-13835520$$

Add 36900c to -36900c. Terms 36900c and -36900c cancel out, leaving an equation with only one variable that can be solved.

$$16810000m-17953200m=13382400-13835520$$

Add 16810000m to -17953200m.

$$-1143200m=13382400-13835520$$

Add 13382400 to -13835520.

$$-1143200m=-453120$$

Divide both sides by -1143200.

$$m=\frac{2832}{7145}$$

Substitute \frac{2832}{7145} for m in 4100c+1994800m=1537280. Because the resulting equation contains only one variable, you can solve for c directly.

$$4100c+1994800\times \frac{2832}{7145}=1537280$$

Multiply 1994800 times \frac{2832}{7145}.

$$4100c+\frac{1129854720}{1429}=1537280$$

Subtract \frac{1129854720}{1429} from both sides of the equation.

$$4100c=\frac{1066918400}{1429}$$

Divide both sides by 4100.

$$c=\frac{260224}{1429}$$

The system is now solved.

$$c=\frac{260224}{1429},m=\frac{2832}{7145}$$