Solve an=a_{1}(n-1)d

$$a n = a _ { 1 } ( n - 1 ) d$$

Solve for a

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

Steps for Solving Linear Equation

Use the distributive property to multiply a_{1} by n-1.

$$an=\left(a_{1}n-a_{1}\right)d$$

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

$$an=a_{1}nd-a_{1}d$$

The equation is in standard form.

$$na=a_{1}dn-a_{1}d$$

Divide both sides by n.

$$\frac{na}{n}=\frac{a_{1}d\left(n-1\right)}{n}$$

Dividing by n undoes the multiplication by n.

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

Solve for a_1

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

Steps for Solving Linear Equation

Use the distributive property to multiply a_{1} by n-1.

$$an=\left(a_{1}n-a_{1}\right)d$$

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

$$an=a_{1}nd-a_{1}d$$

Swap sides so that all variable terms are on the left hand side.

$$a_{1}nd-a_{1}d=an$$

Combine all terms containing a_{1}.

$$\left(nd-d\right)a_{1}=an$$

The equation is in standard form.

$$\left(dn-d\right)a_{1}=an$$

Divide both sides by dn-d.

$$\frac{\left(dn-d\right)a_{1}}{dn-d}=\frac{an}{dn-d}$$

Dividing by dn-d undoes the multiplication by dn-d.

$$a_{1}=\frac{an}{dn-d}$$

Divide an by dn-d.

$$a_{1}=\frac{an}{d\left(n-1\right)}$$