Solve 2an+n(n-1)/2x(-3)=70

$$2 a n + \frac { n ( n - 1 ) } { 2 } x ( - 3 ) = 70$$

Solve for a

$$a=\frac{3nx}{4}-\frac{3x}{4}+\frac{35}{n}$$

$$n\neq 0$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 2.

$$4an+n\left(n-1\right)x\left(-3\right)=140$$

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

$$4an+\left(n^{2}-n\right)x\left(-3\right)=140$$

Use the distributive property to multiply n^{2}-n by x.

$$4an+\left(n^{2}x-nx\right)\left(-3\right)=140$$

Use the distributive property to multiply n^{2}x-nx by -3.

$$4an-3n^{2}x+3nx=140$$

Add 3n^{2}x to both sides.

$$4an+3nx=140+3n^{2}x$$

Subtract 3nx from both sides.

$$4an=140+3n^{2}x-3nx$$

The equation is in standard form.

$$4na=3xn^{2}-3nx+140$$

Divide both sides by 4n.

$$\frac{4na}{4n}=\frac{3xn^{2}-3nx+140}{4n}$$

Dividing by 4n undoes the multiplication by 4n.

$$a=\frac{3xn^{2}-3nx+140}{4n}$$

Divide 140+3n^{2}x-3nx by 4n.

$$a=\frac{3nx}{4}-\frac{3x}{4}+\frac{35}{n}$$

Solve for n (complex solution)

$$\left\{\begin{matrix}n=-\frac{\sqrt{9x^{2}+24ax-1680x+16a^{2}}-3x-4a}{6x}\text{; }n=\frac{\sqrt{9x^{2}+24ax-1680x+16a^{2}}+3x+4a}{6x}\text{, }&x\neq 0\\n=\frac{35}{a}\text{, }&x=0\text{ and }a\neq 0\end{matrix}\right.$$

Solve for n

$$\left\{\begin{matrix}n=-\frac{\sqrt{9x^{2}+24ax-1680x+16a^{2}}-3x-4a}{6x}\text{; }n=\frac{\sqrt{9x^{2}+24ax-1680x+16a^{2}}+3x+4a}{6x}\text{, }&\left(x\neq 0\text{ and }a\geq -\frac{3x}{4}+\sqrt{105x}\right)\text{ or }\left(x\neq 0\text{ and }a\leq -\frac{3x}{4}-\sqrt{105x}\right)\text{ or }x<0\\n=\frac{35}{a}\text{, }&x=0\text{ and }a\neq 0\end{matrix}\right.$$