Solve m/2(2a+(m-1)d)=n/2[2a+(n-1)d]

$$\frac { m } { 2 } ( 2 a + ( m - 1 ) d ) = \frac { n } { 2 } [ 2 a + ( n - 1 ) d ]$$

Solve for a

$$\left\{\begin{matrix}\\a=-\frac{d\left(m+n-1\right)}{2}\text{, }&\text{unconditionally}\\a\in \mathrm{R}\text{, }&m=n\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 2.

$$m\left(2a+\left(m-1\right)d\right)=n\left(2a+\left(n-1\right)d\right)$$

Use the distributive property to multiply m-1 by d.

$$m\left(2a+md-d\right)=n\left(2a+\left(n-1\right)d\right)$$

Use the distributive property to multiply m by 2a+md-d.

$$2ma+dm^{2}-md=n\left(2a+\left(n-1\right)d\right)$$

Use the distributive property to multiply n-1 by d.

$$2ma+dm^{2}-md=n\left(2a+nd-d\right)$$

Use the distributive property to multiply n by 2a+nd-d.

$$2ma+dm^{2}-md=2na+dn^{2}-nd$$

Subtract 2na from both sides.

$$2ma+dm^{2}-md-2na=dn^{2}-nd$$

Subtract dm^{2} from both sides.

$$2ma-md-2na=dn^{2}-nd-dm^{2}$$

Add md to both sides.

$$2ma-2na=dn^{2}-nd-dm^{2}+md$$

Reorder the terms.

$$2am-2an=-dm^{2}+dm+dn^{2}-dn$$

Combine all terms containing a.

$$\left(2m-2n\right)a=-dm^{2}+dm+dn^{2}-dn$$

Divide both sides by 2m-2n.

$$\frac{\left(2m-2n\right)a}{2m-2n}=\frac{d\left(n-m\right)\left(m+n-1\right)}{2m-2n}$$

Dividing by 2m-2n undoes the multiplication by 2m-2n.

$$a=\frac{d\left(n-m\right)\left(m+n-1\right)}{2m-2n}$$

Divide d\left(-1+m+n\right)\left(-m+n\right) by 2m-2n.

$$a=-\frac{d\left(m+n-1\right)}{2}$$

Solve for d

$$\left\{\begin{matrix}d=-\frac{2a}{m+n-1}\text{, }&m\neq 1-n\\d\in \mathrm{R}\text{, }&m=n\text{ or }\left(a=0\text{ and }m=1-n\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 2.

$$m\left(2a+\left(m-1\right)d\right)=n\left(2a+\left(n-1\right)d\right)$$

Use the distributive property to multiply m-1 by d.

$$m\left(2a+md-d\right)=n\left(2a+\left(n-1\right)d\right)$$

Use the distributive property to multiply m by 2a+md-d.

$$2ma+dm^{2}-md=n\left(2a+\left(n-1\right)d\right)$$

Use the distributive property to multiply n-1 by d.

$$2ma+dm^{2}-md=n\left(2a+nd-d\right)$$

Use the distributive property to multiply n by 2a+nd-d.

$$2ma+dm^{2}-md=2na+dn^{2}-nd$$

Subtract dn^{2} from both sides.

$$2ma+dm^{2}-md-dn^{2}=2na-nd$$

Add nd to both sides.

$$2ma+dm^{2}-md-dn^{2}+nd=2na$$

Subtract 2ma from both sides.

$$dm^{2}-md-dn^{2}+nd=2na-2ma$$

Combine all terms containing d.

$$\left(m^{2}-m-n^{2}+n\right)d=2na-2ma$$

The equation is in standard form.

$$\left(m^{2}-m-n^{2}+n\right)d=2an-2am$$

Divide both sides by m^{2}-m-n^{2}+n.

$$\frac{\left(m^{2}-m-n^{2}+n\right)d}{m^{2}-m-n^{2}+n}=\frac{2a\left(n-m\right)}{m^{2}-m-n^{2}+n}$$

Dividing by m^{2}-m-n^{2}+n undoes the multiplication by m^{2}-m-n^{2}+n.

$$d=\frac{2a\left(n-m\right)}{m^{2}-m-n^{2}+n}$$

Divide 2a\left(n-m\right) by m^{2}-m-n^{2}+n.

$$d=-\frac{2a}{m+n-1}$$