Solving Linear Systems: Hemlock & Blue Spruce Trees

[jojonea]

Had this in the wrong forum

I have a question on my homework, it is on Applications of Linear Systems, the question is : You plant a 14-inch hemlock tree in your backyard that grows at a rate of 4 inches per year and an 8-inch blue spruce tree that grows at a rate of 6 inches per year. In how many years after you plant the trees will the two trees be the same height? how tall will each tree be?

I know the answer: 3 years, both 26 inches, I don't know how to write out the systems, maybe I'm just an idiot, I don't know.

 

[MikeH]

Make an equation for each tree in the form of y=mx+b.
Use a let statement so you know what each variable represents.
So your let statement would be:
Let y = total height of tree
Let x = number of years
Let b = initial height
Let m = the rate at which it grows

From here you write your equations substituting the values you know.

For the first tree.
y=4x+14
And the second.
y=6x+8

Then to find x you simply make the 2 equations equal since they are both equal to y.

4x+14 = 6x+8
Solve for x.

Then input your x value into one of your equations and you have your y value.

 

[jojonea]

i feel so stupid...

I was working on that for a whole hour, thank you so much... I didnt realize it was that easy!

 

[MikeH]

Any question of this sort should be done using algebra. If you find yourself using the trial and error method, remember algebra. Besides trial and error gets you nowheres on a test or assignment.

 

[HallsofIvy]

I surely don't want to "dis" algebra but the way I would do this is:

Since the spruce grows 6 inches per year and the hemlock on 4 inches per year, the spruce "catches up" two inches per year. The hemlock was orginally 14 inches and the spruce 8 inches so the spruce needs to catch up 14- 8= 6 inches. At two inches a year, that will take
6/2= 3years. Of course, in three years, the hemlock will have grown 4(3)= 12 inches to a total of 14+12= 26 inches and the spruce will have grown 6(3)= 18 inches, also to a total of 8+18= 26 inches.

 

[MikeH]

That wouldn't be showing your linear equations. All of my teachers gave points on tests and quizzes just for writing the equations and the let statements. By doing that it shows that you know the concept of linear equations so if you make a simple addition mistake or something like that you still get near full value.