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