Solve {l}{m+n/3+n-m/4=-1/4}{m+8/6-frac{5(n+1)}{12}=2}

$$\left\{ \begin{array} { l } { \frac { m + n } { 3 } + \frac { n - m } { 4 } = - \frac { 1 } { 4 } } \\ { \frac { m + 8 } { 6 } - \frac { 5 ( n + 1 ) } { 12 } = 2 } \end{array} \right.$$

Solve for m, n

$$m=4$$

$$n=-1$$

Steps Using Substitution

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

$$4\left(m+n\right)+3\left(n-m\right)=-3$$

Use the distributive property to multiply 4 by m+n.

$$4m+4n+3\left(n-m\right)=-3$$

Use the distributive property to multiply 3 by n-m.

$$4m+4n+3n-3m=-3$$

Combine 4n and 3n to get 7n.

$$4m+7n-3m=-3$$

Combine 4m and -3m to get m.

$$m+7n=-3$$

Consider the second equation. Multiply both sides of the equation by 12, the least common multiple of 6,12.

$$2\left(m+8\right)-5\left(n+1\right)=24$$

Use the distributive property to multiply 2 by m+8.

$$2m+16-5\left(n+1\right)=24$$

Use the distributive property to multiply -5 by n+1.

$$2m+16-5n-5=24$$

Subtract 5 from 16 to get 11.

$$2m+11-5n=24$$

Subtract 11 from both sides.

$$2m-5n=24-11$$

Subtract 11 from 24 to get 13.

$$2m-5n=13$$

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.

$$m+7n=-3,2m-5n=13$$

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

$$m+7n=-3$$

Subtract 7n from both sides of the equation.

$$m=-7n-3$$

Substitute -7n-3 for m in the other equation, 2m-5n=13.

$$2\left(-7n-3\right)-5n=13$$

Multiply 2 times -7n-3.

$$-14n-6-5n=13$$

Add -14n to -5n.

$$-19n-6=13$$

Add 6 to both sides of the equation.

$$-19n=19$$

Divide both sides by -19.

$$n=-1$$

Substitute -1 for n in m=-7n-3. Because the resulting equation contains only one variable, you can solve for m directly.

$$m=-7\left(-1\right)-3$$

Multiply -7 times -1.

$$m=7-3$$

Add -3 to 7.

$$m=4$$

The system is now solved.

$$m=4,n=-1$$

Steps Using Matrices

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

$$4\left(m+n\right)+3\left(n-m\right)=-3$$

Use the distributive property to multiply 4 by m+n.

$$4m+4n+3\left(n-m\right)=-3$$

Use the distributive property to multiply 3 by n-m.

$$4m+4n+3n-3m=-3$$

Combine 4n and 3n to get 7n.

$$4m+7n-3m=-3$$

Combine 4m and -3m to get m.

$$m+7n=-3$$

Consider the second equation. Multiply both sides of the equation by 12, the least common multiple of 6,12.

$$2\left(m+8\right)-5\left(n+1\right)=24$$

Use the distributive property to multiply 2 by m+8.

$$2m+16-5\left(n+1\right)=24$$

Use the distributive property to multiply -5 by n+1.

$$2m+16-5n-5=24$$

Subtract 5 from 16 to get 11.

$$2m+11-5n=24$$

Subtract 11 from both sides.

$$2m-5n=24-11$$

Subtract 11 from 24 to get 13.

$$2m-5n=13$$

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

$$m+7n=-3,2m-5n=13$$

Write the equations in matrix form.

$$\left(\begin{matrix}1&7\\2&-5\end{matrix}\right)\left(\begin{matrix}m\\n\end{matrix}\right)=\left(\begin{matrix}-3\\13\end{matrix}\right)$$

Left multiply the equation by the inverse matrix of \left(\begin{matrix}1&7\\2&-5\end{matrix}\right).

$$inverse(\left(\begin{matrix}1&7\\2&-5\end{matrix}\right))\left(\begin{matrix}1&7\\2&-5\end{matrix}\right)\left(\begin{matrix}m\\n\end{matrix}\right)=inverse(\left(\begin{matrix}1&7\\2&-5\end{matrix}\right))\left(\begin{matrix}-3\\13\end{matrix}\right)$$

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

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}m\\n\end{matrix}\right)=inverse(\left(\begin{matrix}1&7\\2&-5\end{matrix}\right))\left(\begin{matrix}-3\\13\end{matrix}\right)$$

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

$$\left(\begin{matrix}m\\n\end{matrix}\right)=inverse(\left(\begin{matrix}1&7\\2&-5\end{matrix}\right))\left(\begin{matrix}-3\\13\end{matrix}\right)$$

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.

$$\left(\begin{matrix}m\\n\end{matrix}\right)=\left(\begin{matrix}-\frac{5}{-5-7\times 2}&-\frac{7}{-5-7\times 2}\\-\frac{2}{-5-7\times 2}&\frac{1}{-5-7\times 2}\end{matrix}\right)\left(\begin{matrix}-3\\13\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}m\\n\end{matrix}\right)=\left(\begin{matrix}\frac{5}{19}&\frac{7}{19}\\\frac{2}{19}&-\frac{1}{19}\end{matrix}\right)\left(\begin{matrix}-3\\13\end{matrix}\right)$$

Multiply the matrices.

$$\left(\begin{matrix}m\\n\end{matrix}\right)=\left(\begin{matrix}\frac{5}{19}\left(-3\right)+\frac{7}{19}\times 13\\\frac{2}{19}\left(-3\right)-\frac{1}{19}\times 13\end{matrix}\right)$$

Do the arithmetic.

$$\left(\begin{matrix}m\\n\end{matrix}\right)=\left(\begin{matrix}4\\-1\end{matrix}\right)$$

Extract the matrix elements m and n.

$$m=4,n=-1$$

Steps Using Elimination

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

$$4\left(m+n\right)+3\left(n-m\right)=-3$$

Use the distributive property to multiply 4 by m+n.

$$4m+4n+3\left(n-m\right)=-3$$

Use the distributive property to multiply 3 by n-m.

$$4m+4n+3n-3m=-3$$

Combine 4n and 3n to get 7n.

$$4m+7n-3m=-3$$

Combine 4m and -3m to get m.

$$m+7n=-3$$

Consider the second equation. Multiply both sides of the equation by 12, the least common multiple of 6,12.

$$2\left(m+8\right)-5\left(n+1\right)=24$$

Use the distributive property to multiply 2 by m+8.

$$2m+16-5\left(n+1\right)=24$$

Use the distributive property to multiply -5 by n+1.

$$2m+16-5n-5=24$$

Subtract 5 from 16 to get 11.

$$2m+11-5n=24$$

Subtract 11 from both sides.

$$2m-5n=24-11$$

Subtract 11 from 24 to get 13.

$$2m-5n=13$$

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.

$$m+7n=-3,2m-5n=13$$

To make m and 2m equal, multiply all terms on each side of the first equation by 2 and all terms on each side of the second by 1.

$$2m+2\times 7n=2\left(-3\right),2m-5n=13$$

Simplify.

$$2m+14n=-6,2m-5n=13$$

Subtract 2m-5n=13 from 2m+14n=-6 by subtracting like terms on each side of the equal sign.

$$2m-2m+14n+5n=-6-13$$

Add 2m to -2m. Terms 2m and -2m cancel out, leaving an equation with only one variable that can be solved.

$$14n+5n=-6-13$$

Add 14n to 5n.

$$19n=-6-13$$

Add -6 to -13.

$$19n=-19$$

Divide both sides by 19.

$$n=-1$$

Substitute -1 for n in 2m-5n=13. Because the resulting equation contains only one variable, you can solve for m directly.

$$2m-5\left(-1\right)=13$$

Multiply -5 times -1.

$$2m+5=13$$

Subtract 5 from both sides of the equation.

$$2m=8$$

Divide both sides by 2.

$$m=4$$

The system is now solved.

$$m=4,n=-1$$