Solve (m-n)a+[(m+n)(m-n)-(m-n)]d=0

$$( m - n ) a + [ ( m + n ) ( m - n ) - ( m - n ) ] d = 0$$

Solve for a

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

Steps for Solving Linear Equation

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

$$ma-na+\left(\left(m+n\right)\left(m-n\right)-\left(m-n\right)\right)d=0$$

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}.

$$ma-na+\left(m^{2}-n^{2}-\left(m-n\right)\right)d=0$$

To find the opposite of m-n, find the opposite of each term.

$$ma-na+\left(m^{2}-n^{2}-m+n\right)d=0$$

Use the distributive property to multiply m^{2}-n^{2}-m+n by d.

$$ma-na+m^{2}d-n^{2}d-md+nd=0$$

Subtract m^{2}d from both sides. Anything subtracted from zero gives its negation.

$$ma-na-n^{2}d-md+nd=-m^{2}d$$

Add n^{2}d to both sides.

$$ma-na-md+nd=-m^{2}d+n^{2}d$$

Add md to both sides.

$$ma-na+nd=-m^{2}d+n^{2}d+md$$

Subtract nd from both sides.

$$ma-na=-m^{2}d+n^{2}d+md-nd$$

Reorder the terms.

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

Combine all terms containing a.

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

Divide both sides by m-n.

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

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

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

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

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

Solve for d

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

Steps for Solving Linear Equation

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

$$ma-na+\left(\left(m+n\right)\left(m-n\right)-\left(m-n\right)\right)d=0$$

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}.

$$ma-na+\left(m^{2}-n^{2}-\left(m-n\right)\right)d=0$$

To find the opposite of m-n, find the opposite of each term.

$$ma-na+\left(m^{2}-n^{2}-m+n\right)d=0$$

Use the distributive property to multiply m^{2}-n^{2}-m+n by d.

$$ma-na+m^{2}d-n^{2}d-md+nd=0$$

Subtract ma from both sides. Anything subtracted from zero gives its negation.

$$-na+m^{2}d-n^{2}d-md+nd=-ma$$

Add na to both sides.

$$m^{2}d-n^{2}d-md+nd=-ma+na$$

Reorder the terms.

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

Combine all terms containing d.

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

The equation is in standard form.

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

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

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

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

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

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

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