Solve {l}{m/3-n/4=3}{m/2-n/3=13}

$$\left\{ \begin{array} { l } { \frac { m } { 3 } - \frac { n } { 4 } = 3 } \\ { \frac { m } { 2 } - \frac { n } { 3 } = 13 } \end{array} \right.$$

Solve for m, n

$$m=162$$

$$n=204$$

Steps Using Substitution

Consider the first equation. Multiply both sides of the equation by 12, the least common multiple of 3,4.

$$4m-3n=36$$

Consider the second equation. Multiply both sides of the equation by 6, the least common multiple of 2,3.

$$3m-2n=78$$

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.

$$4m-3n=36,3m-2n=78$$

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

$$4m-3n=36$$

Add 3n to both sides of the equation.

$$4m=3n+36$$

Divide both sides by 4.

$$m=\frac{1}{4}\left(3n+36\right)$$

Multiply \frac{1}{4} times 36+3n.

$$m=\frac{3}{4}n+9$$

Substitute \frac{3n}{4}+9 for m in the other equation, 3m-2n=78.

$$3\left(\frac{3}{4}n+9\right)-2n=78$$

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

$$\frac{9}{4}n+27-2n=78$$

Add \frac{9n}{4} to -2n.

$$\frac{1}{4}n+27=78$$

Subtract 27 from both sides of the equation.

$$\frac{1}{4}n=51$$

Multiply both sides by 4.

$$n=204$$

Substitute 204 for n in m=\frac{3}{4}n+9. Because the resulting equation contains only one variable, you can solve for m directly.

$$m=\frac{3}{4}\times 204+9$$

Multiply \frac{3}{4} times 204.

$$m=153+9$$

Add 9 to 153.

$$m=162$$

The system is now solved.

$$m=162,n=204$$

Steps Using Matrices

Consider the first equation. Multiply both sides of the equation by 12, the least common multiple of 3,4.

$$4m-3n=36$$

Consider the second equation. Multiply both sides of the equation by 6, the least common multiple of 2,3.

$$3m-2n=78$$

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

$$4m-3n=36,3m-2n=78$$

Write the equations in matrix form.

$$\left(\begin{matrix}4&-3\\3&-2\end{matrix}\right)\left(\begin{matrix}m\\n\end{matrix}\right)=\left(\begin{matrix}36\\78\end{matrix}\right)$$

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

$$inverse(\left(\begin{matrix}4&-3\\3&-2\end{matrix}\right))\left(\begin{matrix}4&-3\\3&-2\end{matrix}\right)\left(\begin{matrix}m\\n\end{matrix}\right)=inverse(\left(\begin{matrix}4&-3\\3&-2\end{matrix}\right))\left(\begin{matrix}36\\78\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}m\\n\end{matrix}\right)=inverse(\left(\begin{matrix}4&-3\\3&-2\end{matrix}\right))\left(\begin{matrix}36\\78\end{matrix}\right)$$

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

$$\left(\begin{matrix}m\\n\end{matrix}\right)=inverse(\left(\begin{matrix}4&-3\\3&-2\end{matrix}\right))\left(\begin{matrix}36\\78\end{matrix}\right)$$

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.

$$\left(\begin{matrix}m\\n\end{matrix}\right)=\left(\begin{matrix}-\frac{2}{4\left(-2\right)-\left(-3\times 3\right)}&-\frac{-3}{4\left(-2\right)-\left(-3\times 3\right)}\\-\frac{3}{4\left(-2\right)-\left(-3\times 3\right)}&\frac{4}{4\left(-2\right)-\left(-3\times 3\right)}\end{matrix}\right)\left(\begin{matrix}36\\78\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}m\\n\end{matrix}\right)=\left(\begin{matrix}-2&3\\-3&4\end{matrix}\right)\left(\begin{matrix}36\\78\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}m\\n\end{matrix}\right)=\left(\begin{matrix}-2\times 36+3\times 78\\-3\times 36+4\times 78\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}m\\n\end{matrix}\right)=\left(\begin{matrix}162\\204\end{matrix}\right)$$

Extract the matrix elements m and n.

$$m=162,n=204$$

Steps Using Elimination

Consider the first equation. Multiply both sides of the equation by 12, the least common multiple of 3,4.

$$4m-3n=36$$

Consider the second equation. Multiply both sides of the equation by 6, the least common multiple of 2,3.

$$3m-2n=78$$

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.

$$4m-3n=36,3m-2n=78$$

To make 4m and 3m equal, multiply all terms on each side of the first equation by 3 and all terms on each side of the second by 4.

$$3\times 4m+3\left(-3\right)n=3\times 36,4\times 3m+4\left(-2\right)n=4\times 78$$

Simplify.

$$12m-9n=108,12m-8n=312$$

Subtract 12m-8n=312 from 12m-9n=108 by subtracting like terms on each side of the equal sign.

$$12m-12m-9n+8n=108-312$$

Add 12m to -12m. Terms 12m and -12m cancel out, leaving an equation with only one variable that can be solved.

$$-9n+8n=108-312$$

Add -9n to 8n.

$$-n=108-312$$

Add 108 to -312.

$$-n=-204$$

Divide both sides by -1.

$$n=204$$

Substitute 204 for n in 3m-2n=78. Because the resulting equation contains only one variable, you can solve for m directly.

$$3m-2\times 204=78$$

Multiply -2 times 204.

$$3m-408=78$$

Add 408 to both sides of the equation.

$$3m=486$$

Divide both sides by 3.

$$m=162$$

The system is now solved.

$$m=162,n=204$$