Solve S=10t-1/2at^2

$$S = 10t- \frac{ 1 }{ 2 } a { t }^{ 2 }$$

Solve for a

$$\left\{\begin{matrix}a=\frac{2\left(10t-S\right)}{t^{2}}\text{, }&t\neq 0\\a\in \mathrm{R}\text{, }&S=0\text{ and }t=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$10t-\frac{1}{2}at^{2}=S$$

Subtract 10t from both sides.

$$-\frac{1}{2}at^{2}=S-10t$$

The equation is in standard form.

$$\left(-\frac{t^{2}}{2}\right)a=S-10t$$

Divide both sides by -\frac{1}{2}t^{2}.

$$\frac{\left(-\frac{t^{2}}{2}\right)a}{-\frac{t^{2}}{2}}=\frac{S-10t}{-\frac{t^{2}}{2}}$$

Dividing by -\frac{1}{2}t^{2} undoes the multiplication by -\frac{1}{2}t^{2}.

$$a=\frac{S-10t}{-\frac{t^{2}}{2}}$$

Divide S-10t by -\frac{1}{2}t^{2}.

$$a=-\frac{2\left(S-10t\right)}{t^{2}}$$

Solve for S

$$S=-\frac{t\left(at-20\right)}{2}$$