###### Example3.1.2

A water tank can hold $$140$$ gallons of water, but it has only $$5$$ gallons of water. A tap was turned on, pouring $$15$$ gallons of water into the tank every minute. After how many minutes will the tank be full? Let's find a pattern first.

 Minutes since Tap Amount of Water in Was Turned on the Tank (in Gallons) $$0$$ $$5$$ $$1$$ $$15\cdot1+5=20$$ $$2$$ $$15\cdot2+5=35$$ $$3$$ $$15\cdot3+5=50$$ $$4$$ $$15\cdot4+5=65$$ $$\vdots$$ $$\vdots$$ $$m$$ $$15m+5$$

We can see that after $$m$$ minutes, the tank has $$15m+5$$ gallons of water. This makes sense since the tap pours $$15m$$ gallons of water into the tank in $$m$$ minutes and it had $$5$$ gallons to start with. To find when the tank will be full (with $$140$$ gallons of water), we can write the equation

\begin{equation*} 15m+5=140 \end{equation*}

First, we need to isolate the variable term, $$15m\text{,}$$ in the equation. In other words, we need to remove $$5$$ from the left side of the equals sign. We can do this by subtracting $$5$$ from both sides of the equation. Once the variable term is isolated, we can eliminate the coefficient and solve for $$m\text{.}$$ The full process is:

\begin{align*} 15m+5\amp=140\\ 15m+5\subtractright{5}\amp=140\subtractright{5}\\ 15m\amp=135\\ \divideunder{15m}{15}\amp=\divideunder{135}{15}\\ m\amp=9 \end{align*}

Next, we need to substitute $$m$$ with $$9$$ in the equation $$15m+5=140$$ to check the solution:

\begin{align*} 15m+5\amp=140\\ 15(\substitute{9})+5\amp\stackrel{?}{=}140\\ 135+5\amp\stackrel{\checkmark}{=}140 \end{align*}

The solution $$9$$ is checked. In summary, the tank will be full after $$9$$ minutes.

in-context