Solve {l}{2m-3n=9}{4m+9n=21}

$$\left. \begin{array} { l } { 2 m - 3 n = 9 } \\ { 4 m + 9 n = 21 } \end{array} \right.$$

Solve for m, n

$$m = \frac{24}{5} = 4\frac{4}{5} = 4.8$$

$$n=\frac{1}{5}=0.2$$

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.

$$2m-3n=9,4m+9n=21$$

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

$$2m-3n=9$$

Add 3n to both sides of the equation.

$$2m=3n+9$$

Divide both sides by 2.

$$m=\frac{1}{2}\left(3n+9\right)$$

Multiply \frac{1}{2} times 9+3n.

$$m=\frac{3}{2}n+\frac{9}{2}$$

Substitute \frac{9+3n}{2} for m in the other equation, 4m+9n=21.

$$4\left(\frac{3}{2}n+\frac{9}{2}\right)+9n=21$$

Multiply 4 times \frac{9+3n}{2}.

$$6n+18+9n=21$$

Add 6n to 9n.

$$15n+18=21$$

Subtract 18 from both sides of the equation.

$$15n=3$$

Divide both sides by 15.

$$n=\frac{1}{5}$$

Substitute \frac{1}{5} for n in m=\frac{3}{2}n+\frac{9}{2}. Because the resulting equation contains only one variable, you can solve for m directly.

$$m=\frac{3}{2}\times \frac{1}{5}+\frac{9}{2}$$

Multiply \frac{3}{2} times \frac{1}{5} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$m=\frac{3}{10}+\frac{9}{2}$$

Add \frac{9}{2} to \frac{3}{10} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$m=\frac{24}{5}$$

The system is now solved.

$$m=\frac{24}{5},n=\frac{1}{5}$$

Steps Using Matrices

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

$$2m-3n=9,4m+9n=21$$

Write the equations in matrix form.

$$\left(\begin{matrix}2&-3\\4&9\end{matrix}\right)\left(\begin{matrix}m\\n\end{matrix}\right)=\left(\begin{matrix}9\\21\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}2&-3\\4&9\end{matrix}\right).

$$inverse(\left(\begin{matrix}2&-3\\4&9\end{matrix}\right))\left(\begin{matrix}2&-3\\4&9\end{matrix}\right)\left(\begin{matrix}m\\n\end{matrix}\right)=inverse(\left(\begin{matrix}2&-3\\4&9\end{matrix}\right))\left(\begin{matrix}9\\21\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}m\\n\end{matrix}\right)=inverse(\left(\begin{matrix}2&-3\\4&9\end{matrix}\right))\left(\begin{matrix}9\\21\end{matrix}\right)$$

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

$$\left(\begin{matrix}m\\n\end{matrix}\right)=inverse(\left(\begin{matrix}2&-3\\4&9\end{matrix}\right))\left(\begin{matrix}9\\21\end{matrix}\right)$$

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.

$$\left(\begin{matrix}m\\n\end{matrix}\right)=\left(\begin{matrix}\frac{9}{2\times 9-\left(-3\times 4\right)}&-\frac{-3}{2\times 9-\left(-3\times 4\right)}\\-\frac{4}{2\times 9-\left(-3\times 4\right)}&\frac{2}{2\times 9-\left(-3\times 4\right)}\end{matrix}\right)\left(\begin{matrix}9\\21\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}m\\n\end{matrix}\right)=\left(\begin{matrix}\frac{3}{10}&\frac{1}{10}\\-\frac{2}{15}&\frac{1}{15}\end{matrix}\right)\left(\begin{matrix}9\\21\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}m\\n\end{matrix}\right)=\left(\begin{matrix}\frac{3}{10}\times 9+\frac{1}{10}\times 21\\-\frac{2}{15}\times 9+\frac{1}{15}\times 21\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}m\\n\end{matrix}\right)=\left(\begin{matrix}\frac{24}{5}\\\frac{1}{5}\end{matrix}\right)$$

Extract the matrix elements m and n.

$$m=\frac{24}{5},n=\frac{1}{5}$$

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.

$$2m-3n=9,4m+9n=21$$

To make 2m and 4m equal, multiply all terms on each side of the first equation by 4 and all terms on each side of the second by 2.

$$4\times 2m+4\left(-3\right)n=4\times 9,2\times 4m+2\times 9n=2\times 21$$

Simplify.

$$8m-12n=36,8m+18n=42$$

Subtract 8m+18n=42 from 8m-12n=36 by subtracting like terms on each side of the equal sign.

$$8m-8m-12n-18n=36-42$$

Add 8m to -8m. Terms 8m and -8m cancel out, leaving an equation with only one variable that can be solved.

$$-12n-18n=36-42$$

Add -12n to -18n.

$$-30n=36-42$$

Add 36 to -42.

$$-30n=-6$$

Divide both sides by -30.

$$n=\frac{1}{5}$$

Substitute \frac{1}{5} for n in 4m+9n=21. Because the resulting equation contains only one variable, you can solve for m directly.

$$4m+9\times \frac{1}{5}=21$$

Multiply 9 times \frac{1}{5}.

$$4m+\frac{9}{5}=21$$

Subtract \frac{9}{5} from both sides of the equation.

$$4m=\frac{96}{5}$$

Divide both sides by 4.

$$m=\frac{24}{5}$$

The system is now solved.

$$m=\frac{24}{5},n=\frac{1}{5}$$