Example4.8.19

Two trees were planted in the same year, and their growth over time is modeled by the two lines in FigureĀ 4.8.20. Use linear equations to model each tree's growth, and interpret their meanings in this context.

This is a Cartesian grid with two lines. A red line passes (0,2), (3,2), (3,4) ... The points (0,2) and (3,4) are plotted. A slope triangle is drawn from (0,2), passes (3,2) and ends at (3,4). Its base is labeled "3-0=3", and its height is labeled "4-2=2". A blue line passes (0,5), (3,5), (3,7) ... The points (0,5) and (3,7) are plotted. A slope triangle is drawn from (0,5), passes (3,5) and ends at (3,7). Its base is labeled "3-0=3", and its height is labeled "7-5=2".
Figure4.8.20Two Trees' Growth Chart

We can see Tree 1's equation is \(y=\frac{2}{3}x+2\text{,}\) and Tree 2's equation is \(y=\frac{2}{3}+5\text{.}\) Tree 1 was \(2\) feet tall when it was planted, and Tree 2 was \(5\) feet tall when it was planted. Both trees have been growing at the same rate, \(\frac{2}{3}\) feet per year, or \(2\) feet every \(3\) years.

An important observation right now is that those two lines are parallel. Why? For lines with positive slopes, the bigger a line's slope, the steeper the line is slanted. As a result, if two lines have the same slope, they are slanted at the same angle, thus they are parallel.

in-context