Solve {l}{2s+3c=11}{3s+6c=19*50}

$$\left. \begin{array} { l } { 2 s + 3 c = 11 } \\ { 3 s + 6 c = 19 \cdot 50 } \end{array} \right.$$

Solve for s, c

$$s=-928$$

$$c = \frac{1867}{3} = 622\frac{1}{3} \approx 622.333333333$$

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.

$$2s+3c=11,3s+6c=950$$

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

$$2s+3c=11$$

Subtract 3c from both sides of the equation.

$$2s=-3c+11$$

Divide both sides by 2.

$$s=\frac{1}{2}\left(-3c+11\right)$$

Multiply \frac{1}{2} times -3c+11.

$$s=-\frac{3}{2}c+\frac{11}{2}$$

Substitute \frac{-3c+11}{2} for s in the other equation, 3s+6c=950.

$$3\left(-\frac{3}{2}c+\frac{11}{2}\right)+6c=950$$

Multiply 3 times \frac{-3c+11}{2}.

$$-\frac{9}{2}c+\frac{33}{2}+6c=950$$

Add -\frac{9c}{2} to 6c.

$$\frac{3}{2}c+\frac{33}{2}=950$$

Subtract \frac{33}{2} from both sides of the equation.

$$\frac{3}{2}c=\frac{1867}{2}$$

Divide both sides of the equation by \frac{3}{2}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$c=\frac{1867}{3}$$

Substitute \frac{1867}{3} for c in s=-\frac{3}{2}c+\frac{11}{2}. Because the resulting equation contains only one variable, you can solve for s directly.

$$s=-\frac{3}{2}\times \frac{1867}{3}+\frac{11}{2}$$

Multiply -\frac{3}{2} times \frac{1867}{3} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$s=\frac{-1867+11}{2}$$

Add \frac{11}{2} to -\frac{1867}{2} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$s=-928$$

The system is now solved.

$$s=-928,c=\frac{1867}{3}$$

Steps Using Matrices

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

$$2s+3c=11,3s+6c=950$$

Write the equations in matrix form.

$$\left(\begin{matrix}2&3\\3&6\end{matrix}\right)\left(\begin{matrix}s\\c\end{matrix}\right)=\left(\begin{matrix}11\\950\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}2&3\\3&6\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&3\\3&6\end{matrix}\right))\left(\begin{matrix}2&3\\3&6\end{matrix}\right)\left(\begin{matrix}s\\c\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\3&6\end{matrix}\right))\left(\begin{matrix}11\\950\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}s\\c\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\3&6\end{matrix}\right))\left(\begin{matrix}11\\950\end{matrix}\right)$$

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

$$\left(\begin{matrix}s\\c\end{matrix}\right)=inverse(\left(\begin{matrix}2&3\\3&6\end{matrix}\right))\left(\begin{matrix}11\\950\end{matrix}\right)$$

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.

$$\left(\begin{matrix}s\\c\end{matrix}\right)=\left(\begin{matrix}\frac{6}{2\times 6-3\times 3}&-\frac{3}{2\times 6-3\times 3}\\-\frac{3}{2\times 6-3\times 3}&\frac{2}{2\times 6-3\times 3}\end{matrix}\right)\left(\begin{matrix}11\\950\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}s\\c\end{matrix}\right)=\left(\begin{matrix}2&-1\\-1&\frac{2}{3}\end{matrix}\right)\left(\begin{matrix}11\\950\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}s\\c\end{matrix}\right)=\left(\begin{matrix}2\times 11-950\\-11+\frac{2}{3}\times 950\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}s\\c\end{matrix}\right)=\left(\begin{matrix}-928\\\frac{1867}{3}\end{matrix}\right)$$

Extract the matrix elements s and c.

$$s=-928,c=\frac{1867}{3}$$

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.

$$2s+3c=11,3s+6c=950$$

To make 2s and 3s equal, multiply all terms on each side of the first equation by 3 and all terms on each side of the second by 2.

$$3\times 2s+3\times 3c=3\times 11,2\times 3s+2\times 6c=2\times 950$$

Simplify.

$$6s+9c=33,6s+12c=1900$$

Subtract 6s+12c=1900 from 6s+9c=33 by subtracting like terms on each side of the equal sign.

$$6s-6s+9c-12c=33-1900$$

Add 6s to -6s. Terms 6s and -6s cancel out, leaving an equation with only one variable that can be solved.

$$9c-12c=33-1900$$

Add 9c to -12c.

$$-3c=33-1900$$

Add 33 to -1900.

$$-3c=-1867$$

Divide both sides by -3.

$$c=\frac{1867}{3}$$

Substitute \frac{1867}{3} for c in 3s+6c=950. Because the resulting equation contains only one variable, you can solve for s directly.

$$3s+6\times \frac{1867}{3}=950$$

Multiply 6 times \frac{1867}{3}.

$$3s+3734=950$$

Subtract 3734 from both sides of the equation.

$$3s=-2784$$

Divide both sides by 3.

$$s=-928$$

The system is now solved.

$$s=-928,c=\frac{1867}{3}$$