Solve {l}{m+n/2-m-n/3=1}{m+n/3m-n/4=-1}

$$\left\{ \begin{array} { l } { \frac { m + n } { 2 } - \frac { m - n } { 3 } = 1 } \\ { \frac { m + n } { 3 } \frac { m - n } { 4 } = - 1 } \end{array} \right.$$

Solve for m, n

$$m=\frac{-1+5\sqrt{7}i}{4}\approx -0.25+3.307189139i\text{, }n=\frac{-\sqrt{7}i+5}{4}\approx 1.25-0.661437828i$$

$$m=\frac{-5\sqrt{7}i-1}{4}\approx -0.25-3.307189139i\text{, }n=\frac{5+\sqrt{7}i}{4}\approx 1.25+0.661437828i$$

Steps Using Substitution and Quadratic Formula

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

$$3\left(m+n\right)-2\left(m-n\right)=6$$

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

$$3m+3n-2\left(m-n\right)=6$$

Use the distributive property to multiply -2 by m-n.

$$3m+3n-2m+2n=6$$

Combine 3m and -2m to get m.

$$m+3n+2n=6$$

Combine 3n and 2n to get 5n.

$$m+5n=6$$

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

$$\left(m+n\right)\left(m-n\right)=-12$$

Consider \left(m+n\right)\left(m-n\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$m^{2}-n^{2}=-12$$

Solve m+5n=6 for m by isolating m on the left hand side of the equal sign.

$$m+5n=6$$

Subtract 5n from both sides of the equation.

$$m=-5n+6$$

Substitute -5n+6 for m in the other equation, -n^{2}+m^{2}=-12.

$$-n^{2}+\left(-5n+6\right)^{2}=-12$$

Square -5n+6.

$$-n^{2}+25n^{2}-60n+36=-12$$

Add -n^{2} to 25n^{2}.

$$24n^{2}-60n+36=-12$$

Add 12 to both sides of the equation.

$$24n^{2}-60n+48=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1+1\left(-5\right)^{2} for a, 1\times 6\left(-5\right)\times 2 for b, and 48 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-\left(-60\right)±\sqrt{\left(-60\right)^{2}-4\times 24\times 48}}{2\times 24}$$

Square 1\times 6\left(-5\right)\times 2.

$$n=\frac{-\left(-60\right)±\sqrt{3600-4\times 24\times 48}}{2\times 24}$$

Multiply -4 times -1+1\left(-5\right)^{2}.

$$n=\frac{-\left(-60\right)±\sqrt{3600-96\times 48}}{2\times 24}$$

Multiply -96 times 48.

$$n=\frac{-\left(-60\right)±\sqrt{3600-4608}}{2\times 24}$$

Add 3600 to -4608.

$$n=\frac{-\left(-60\right)±\sqrt{-1008}}{2\times 24}$$

Take the square root of -1008.

$$n=\frac{-\left(-60\right)±12\sqrt{7}i}{2\times 24}$$

The opposite of 1\times 6\left(-5\right)\times 2 is 60.

$$n=\frac{60±12\sqrt{7}i}{2\times 24}$$

Multiply 2 times -1+1\left(-5\right)^{2}.

$$n=\frac{60±12\sqrt{7}i}{48}$$

Now solve the equation n=\frac{60±12\sqrt{7}i}{48} when ± is plus. Add 60 to 12i\sqrt{7}.

$$n=\frac{60+12\sqrt{7}i}{48}$$

Divide 60+12i\sqrt{7} by 48.

$$n=\frac{5+\sqrt{7}i}{4}$$

Now solve the equation n=\frac{60±12\sqrt{7}i}{48} when ± is minus. Subtract 12i\sqrt{7} from 60.

$$n=\frac{-12\sqrt{7}i+60}{48}$$

Divide 60-12i\sqrt{7} by 48.

$$n=\frac{-\sqrt{7}i+5}{4}$$

There are two solutions for n: \frac{5+i\sqrt{7}}{4} and \frac{5-i\sqrt{7}}{4}. Substitute \frac{5+i\sqrt{7}}{4} for n in the equation m=-5n+6 to find the corresponding solution for m that satisfies both equations.

$$m=-5\times \frac{5+\sqrt{7}i}{4}+6$$

Now substitute \frac{5-i\sqrt{7}}{4} for n in the equation m=-5n+6 and solve to find the corresponding solution for m that satisfies both equations.

$$m=-5\times \frac{-\sqrt{7}i+5}{4}+6$$

The system is now solved.

$$m=-5\times \frac{5+\sqrt{7}i}{4}+6,n=\frac{5+\sqrt{7}i}{4}\text{ or }m=-5\times \frac{-\sqrt{7}i+5}{4}+6,n=\frac{-\sqrt{7}i+5}{4}$$