Solve {l}{3A+4C=136}{2A+3C=97}

$$\left. \begin{array} { l } { 3 A + 4 C = 136 } \\ { 2 A + 3 C = 97 } \end{array} \right.$$

Solve for A, C

$$A=20$$

$$C=19$$

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.

$$3A+4C=136,2A+3C=97$$

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

$$3A+4C=136$$

Subtract 4C from both sides of the equation.

$$3A=-4C+136$$

Divide both sides by 3.

$$A=\frac{1}{3}\left(-4C+136\right)$$

Multiply \frac{1}{3} times -4C+136.

$$A=-\frac{4}{3}C+\frac{136}{3}$$

Substitute \frac{-4C+136}{3} for A in the other equation, 2A+3C=97.

$$2\left(-\frac{4}{3}C+\frac{136}{3}\right)+3C=97$$

Multiply 2 times \frac{-4C+136}{3}.

$$-\frac{8}{3}C+\frac{272}{3}+3C=97$$

Add -\frac{8C}{3} to 3C.

$$\frac{1}{3}C+\frac{272}{3}=97$$

Subtract \frac{272}{3} from both sides of the equation.

$$\frac{1}{3}C=\frac{19}{3}$$

Multiply both sides by 3.

$$C=19$$

Substitute 19 for C in A=-\frac{4}{3}C+\frac{136}{3}. Because the resulting equation contains only one variable, you can solve for A directly.

$$A=-\frac{4}{3}\times 19+\frac{136}{3}$$

Multiply -\frac{4}{3} times 19.

$$A=\frac{-76+136}{3}$$

Add \frac{136}{3} to -\frac{76}{3} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$A=20$$

The system is now solved.

$$A=20,C=19$$

Steps Using Matrices

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

$$3A+4C=136,2A+3C=97$$

Write the equations in matrix form.

$$\left(\begin{matrix}3&4\\2&3\end{matrix}\right)\left(\begin{matrix}A\\C\end{matrix}\right)=\left(\begin{matrix}136\\97\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}3&4\\2&3\end{matrix}\right).

$$inverse(\left(\begin{matrix}3&4\\2&3\end{matrix}\right))\left(\begin{matrix}3&4\\2&3\end{matrix}\right)\left(\begin{matrix}A\\C\end{matrix}\right)=inverse(\left(\begin{matrix}3&4\\2&3\end{matrix}\right))\left(\begin{matrix}136\\97\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}A\\C\end{matrix}\right)=inverse(\left(\begin{matrix}3&4\\2&3\end{matrix}\right))\left(\begin{matrix}136\\97\end{matrix}\right)$$

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

$$\left(\begin{matrix}A\\C\end{matrix}\right)=inverse(\left(\begin{matrix}3&4\\2&3\end{matrix}\right))\left(\begin{matrix}136\\97\end{matrix}\right)$$

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.

$$\left(\begin{matrix}A\\C\end{matrix}\right)=\left(\begin{matrix}\frac{3}{3\times 3-4\times 2}&-\frac{4}{3\times 3-4\times 2}\\-\frac{2}{3\times 3-4\times 2}&\frac{3}{3\times 3-4\times 2}\end{matrix}\right)\left(\begin{matrix}136\\97\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}A\\C\end{matrix}\right)=\left(\begin{matrix}3&-4\\-2&3\end{matrix}\right)\left(\begin{matrix}136\\97\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}A\\C\end{matrix}\right)=\left(\begin{matrix}3\times 136-4\times 97\\-2\times 136+3\times 97\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}A\\C\end{matrix}\right)=\left(\begin{matrix}20\\19\end{matrix}\right)$$

Extract the matrix elements A and C.

$$A=20,C=19$$

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.

$$3A+4C=136,2A+3C=97$$

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

$$2\times 3A+2\times 4C=2\times 136,3\times 2A+3\times 3C=3\times 97$$

Simplify.

$$6A+8C=272,6A+9C=291$$

Subtract 6A+9C=291 from 6A+8C=272 by subtracting like terms on each side of the equal sign.

$$6A-6A+8C-9C=272-291$$

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

$$8C-9C=272-291$$

Add 8C to -9C.

$$-C=272-291$$

Add 272 to -291.

$$-C=-19$$

Divide both sides by -1.

$$C=19$$

Substitute 19 for C in 2A+3C=97. Because the resulting equation contains only one variable, you can solve for A directly.

$$2A+3\times 19=97$$

Multiply 3 times 19.

$$2A+57=97$$

Subtract 57 from both sides of the equation.

$$2A=40$$

Divide both sides by 2.

$$A=20$$

The system is now solved.

$$A=20,C=19$$