Solve {l}{323=36c+6d}{889=81c+9d}

$$\left. \begin{array} { l } { 323 = 36 c + 6 d } \\ { 889 = 81 c + 9 d } \end{array} \right.$$

Solve for c, d

$$c = \frac{809}{54} = 14\frac{53}{54} \approx 14.981481481$$

$$d = -\frac{649}{18} = -36\frac{1}{18} \approx -36.055555556$$

Steps Using Substitution

Consider the first equation. Swap sides so that all variable terms are on the left hand side.

$$36c+6d=323$$

Consider the second equation. Swap sides so that all variable terms are on the left hand side.

$$81c+9d=889$$

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.

$$36c+6d=323,81c+9d=889$$

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

$$36c+6d=323$$

Subtract 6d from both sides of the equation.

$$36c=-6d+323$$

Divide both sides by 36.

$$c=\frac{1}{36}\left(-6d+323\right)$$

Multiply \frac{1}{36} times -6d+323.

$$c=-\frac{1}{6}d+\frac{323}{36}$$

Substitute -\frac{d}{6}+\frac{323}{36} for c in the other equation, 81c+9d=889.

$$81\left(-\frac{1}{6}d+\frac{323}{36}\right)+9d=889$$

Multiply 81 times -\frac{d}{6}+\frac{323}{36}.

$$-\frac{27}{2}d+\frac{2907}{4}+9d=889$$

Add -\frac{27d}{2} to 9d.

$$-\frac{9}{2}d+\frac{2907}{4}=889$$

Subtract \frac{2907}{4} from both sides of the equation.

$$-\frac{9}{2}d=\frac{649}{4}$$

Divide both sides of the equation by -\frac{9}{2}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$d=-\frac{649}{18}$$

Substitute -\frac{649}{18} for d in c=-\frac{1}{6}d+\frac{323}{36}. Because the resulting equation contains only one variable, you can solve for c directly.

$$c=-\frac{1}{6}\left(-\frac{649}{18}\right)+\frac{323}{36}$$

Multiply -\frac{1}{6} times -\frac{649}{18} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$c=\frac{649}{108}+\frac{323}{36}$$

Add \frac{323}{36} to \frac{649}{108} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$c=\frac{809}{54}$$

The system is now solved.

$$c=\frac{809}{54},d=-\frac{649}{18}$$

Steps Using Matrices

Consider the first equation. Swap sides so that all variable terms are on the left hand side.

$$36c+6d=323$$

Consider the second equation. Swap sides so that all variable terms are on the left hand side.

$$81c+9d=889$$

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

$$36c+6d=323,81c+9d=889$$

Write the equations in matrix form.

$$\left(\begin{matrix}36&6\\81&9\end{matrix}\right)\left(\begin{matrix}c\\d\end{matrix}\right)=\left(\begin{matrix}323\\889\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}36&6\\81&9\end{matrix}\right).

$$inverse(\left(\begin{matrix}36&6\\81&9\end{matrix}\right))\left(\begin{matrix}36&6\\81&9\end{matrix}\right)\left(\begin{matrix}c\\d\end{matrix}\right)=inverse(\left(\begin{matrix}36&6\\81&9\end{matrix}\right))\left(\begin{matrix}323\\889\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}c\\d\end{matrix}\right)=inverse(\left(\begin{matrix}36&6\\81&9\end{matrix}\right))\left(\begin{matrix}323\\889\end{matrix}\right)$$

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

$$\left(\begin{matrix}c\\d\end{matrix}\right)=inverse(\left(\begin{matrix}36&6\\81&9\end{matrix}\right))\left(\begin{matrix}323\\889\end{matrix}\right)$$

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.

$$\left(\begin{matrix}c\\d\end{matrix}\right)=\left(\begin{matrix}\frac{9}{36\times 9-6\times 81}&-\frac{6}{36\times 9-6\times 81}\\-\frac{81}{36\times 9-6\times 81}&\frac{36}{36\times 9-6\times 81}\end{matrix}\right)\left(\begin{matrix}323\\889\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}c\\d\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{18}&\frac{1}{27}\\\frac{1}{2}&-\frac{2}{9}\end{matrix}\right)\left(\begin{matrix}323\\889\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}c\\d\end{matrix}\right)=\left(\begin{matrix}-\frac{1}{18}\times 323+\frac{1}{27}\times 889\\\frac{1}{2}\times 323-\frac{2}{9}\times 889\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}c\\d\end{matrix}\right)=\left(\begin{matrix}\frac{809}{54}\\-\frac{649}{18}\end{matrix}\right)$$

Extract the matrix elements c and d.

$$c=\frac{809}{54},d=-\frac{649}{18}$$

Steps Using Elimination

Consider the first equation. Swap sides so that all variable terms are on the left hand side.

$$36c+6d=323$$

Consider the second equation. Swap sides so that all variable terms are on the left hand side.

$$81c+9d=889$$

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.

$$36c+6d=323,81c+9d=889$$

To make 36c and 81c equal, multiply all terms on each side of the first equation by 81 and all terms on each side of the second by 36.

$$81\times 36c+81\times 6d=81\times 323,36\times 81c+36\times 9d=36\times 889$$

Simplify.

$$2916c+486d=26163,2916c+324d=32004$$

Subtract 2916c+324d=32004 from 2916c+486d=26163 by subtracting like terms on each side of the equal sign.

$$2916c-2916c+486d-324d=26163-32004$$

Add 2916c to -2916c. Terms 2916c and -2916c cancel out, leaving an equation with only one variable that can be solved.

$$486d-324d=26163-32004$$

Add 486d to -324d.

$$162d=26163-32004$$

Add 26163 to -32004.

$$162d=-5841$$

Divide both sides by 162.

$$d=-\frac{649}{18}$$

Substitute -\frac{649}{18} for d in 81c+9d=889. Because the resulting equation contains only one variable, you can solve for c directly.

$$81c+9\left(-\frac{649}{18}\right)=889$$

Multiply 9 times -\frac{649}{18}.

$$81c-\frac{649}{2}=889$$

Add \frac{649}{2} to both sides of the equation.

$$81c=\frac{2427}{2}$$

Divide both sides by 81.

$$c=\frac{809}{54}$$

The system is now solved.

$$c=\frac{809}{54},d=-\frac{649}{18}$$