Solve W=-1/2t^2+(14+2a)t-96a+48

$$W = - \frac { 1 } { 2 } t ^ { 2 } + ( 14 + 2 a ) t - 96 a + 48$$

Solve for a

$$\left\{\begin{matrix}a=-\frac{96-2W+28t-t^{2}}{4\left(t-48\right)}\text{, }&t\neq 48\\a\in \mathrm{R}\text{, }&W=-432\text{ and }t=48\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply 14+2a by t.

$$W=-\frac{1}{2}t^{2}+14t+2at-96a+48$$

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

$$-\frac{1}{2}t^{2}+14t+2at-96a+48=W$$

Add \frac{1}{2}t^{2} to both sides.

$$14t+2at-96a+48=W+\frac{1}{2}t^{2}$$

Subtract 14t from both sides.

$$2at-96a+48=W+\frac{1}{2}t^{2}-14t$$

Subtract 48 from both sides.

$$2at-96a=W+\frac{1}{2}t^{2}-14t-48$$

Combine all terms containing a.

$$\left(2t-96\right)a=W+\frac{1}{2}t^{2}-14t-48$$

The equation is in standard form.

$$\left(2t-96\right)a=\frac{t^{2}}{2}+W-14t-48$$

Divide both sides by 2t-96.

$$\frac{\left(2t-96\right)a}{2t-96}=\frac{\frac{t^{2}}{2}+W-14t-48}{2t-96}$$

Dividing by 2t-96 undoes the multiplication by 2t-96.

$$a=\frac{\frac{t^{2}}{2}+W-14t-48}{2t-96}$$

Divide W+\frac{t^{2}}{2}-14t-48 by 2t-96.

$$a=\frac{t^{2}-28t+2W-96}{4\left(t-48\right)}$$

Solve for W

$$W=2at-\frac{t^{2}}{2}+14t-96a+48$$

Solution Steps

Use the distributive property to multiply 14+2a by t.

$$W=-\frac{1}{2}t^{2}+14t+2at-96a+48$$