###### Example4.7.5

A restaurant purchased 1200 lb of flour. The restaurant uses about $$32$$ lb of flour every day. Model the amount of flour that remains $$x$$ days later with a linear equation, and interpret the meaning of its $$x$$-intercept and $$y$$-intercept.

Since the rate of change is constant (-32 lb every day), and we know the initial value, we can model the amount of flour at this restaurant with a slope-intercept equation (4.5.1):

\begin{equation*} y=-32x+1200 \end{equation*}

where $$x$$ represents the number of days passed since the initial purchase, and $$y$$ represents the amount of flour left (in lb.)

A line's $$x$$-intercept is in the form of $$(x,0)\text{,}$$ since to be on the $$x$$-axis, the $$y$$-coordinate must be $$0\text{.}$$ To find this line's $$x$$-intercept, we substitute $$y$$ in the equation with $$0\text{,}$$ and solve for $$x\text{:}$$

\begin{align*} y\amp=-32x+1200\\ \substitute{0}\amp=-32x+1200\\ 0\subtractright{1200}\amp=-32x\\ -1200\amp=-32x\\ \divideunder{-1200}{-32}\amp=x\\ 37.5\amp=x \end{align*}

So the line's $$x$$-intercept is at $$(37.5,0)\text{.}$$ In context this means that $$37.5$$ days after the 1200 lb of flour was purchased, all the flour would be used up.

A line's $$y$$-intercept is in the form of $$(0,y)\text{.}$$ This line equation is already in slope-intercept form, so we can just see that its $$y$$-intercept is at $$(0,1200)\text{.}$$ In general though, we would substitute $$x$$ in the equation with $$0\text{,}$$ and we have:

\begin{align*} y\amp=-32x+1200\\ y\amp=-32(\substitute{0})+1200\\ y\amp=1200 \end{align*}

So yes, the line's $$y$$-intercept is at $$(0,1200)\text{.}$$ This means that on the day the flour was purchased, there was 1200 lb of it. In other words, the $$y$$-intercept tells us one of the original pieces of information: in the beginning, the restaurant purchased 1200 lb of flour.

in-context