Brittany buys two trees for her backyard, one Oak tree and one Pine tree. She plants them appropriately and measures their initial height. Every year Brittany measures the height of the two trees and
eventually realizes that the two trees must have had the same height at some point! Construct two linear equations in the form y = mx + b from the table and determine when the trees were the
same height.
Year
0
Oak
2
1
5
2
8
Pine
6
8
10
10
Let the initial height of the Oak tree be b1 and the initial height of the Pine tree be b2.
For the Oak tree, we have the following data points:
(0, b1)
(1, b1 + m1)
(2, b1 + 2m1)
For the Pine tree, we have the following data points:
(0, b2)
(1, b2 + m2)
(2, b2 + 2m2)
We can construct two linear equations as follows:
For the Oak tree:
y1 = m1x + b1
From the data points:
b1 = 0
b1 + m1 = 2
b1 + 2m1 = 5
Solving these equations, we get:
m1 = 2
b1 = 0
So, the equation for the height of the Oak tree is:
y1 = 2x
For the Pine tree:
y2 = m2x + b2
From the data points:
b2 = 0
b2 + m2 = 6
b2 + 2m2 = 8
Solving these equations, we get:
m2 = 3
b2 = 0
So, the equation for the height of the Pine tree is:
y2 = 3x
Now, to determine when the two trees were the same height, we need to find when y1 = y2:
2x = 3x
x = 0
Therefore, the two trees were the same height at Year 0.