Solving Linear Systems: 14" Hemlock & 8" Blue Spruce

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In summary, the hemlock tree will be 3 years taller than the spruce tree after they are initially the same height.
  • #1
jojonea
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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.
 
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  • #2
I'll try to help, but keep in mind - I don't have linear algebra until next year due to some curriculum changes - so I'll be working mostly from high school knowledge.

Basically, you should try to find an equation that represents the height of each tree. For example, let H be the height of the hemlock tree, and let S be the height of the spruce tree.

Let t represent the number of years that have passed.

Now, you just have to try and find a way to relate H and t and S and t. That is, find H and S as a function of t.

H = H(t)
S = S(t)

Then, set H and S equal to each other, and solve for t.

Keep in mind that when you're solving this problem, the t for both trees is the same as well as the height.

Does that help?
 
  • #3
Hardly a matter of Linear Algebra!

The hemlock is initially 14 in and increases 4 in every year:
After 1 year 14+4 inches,after 2 years 14+ 4+ 4= 14+ 4(2), after 3 years, 14+ 4+ 4+ 4= 14+4(3), etc. Taking H to be the height of the hemlock and t the number of years, H= 14+ 4t.

The spruce is initially 8 inches and grows 6 inches each year: taking S to be the height of the spruce and t the number of years,
S= 8+ 6t.

They will be "equal height" when H= S. That is, when 14+ 4t= 8+ 6t.
Solve for t and then find the height for that t.

That's fairly basic algebra.

You could also do this by noting that, since the spruce grows 6 inches each year, while the hemlock grows only 4 inches, the spruce "catches up" 2 in per year. Since the hemlock is originally 14- 8= 6 inches higher, it will take the spruce 6/2= 3 years to catch up to the hemlock.
 
  • #4
A little help

To make answring this questions easier, you can just create a formula
(equation more accurately).

you have the following "knowns":
1- plants height 14, 8 inches
2- plants rate of growth 4(hemlock) and 6(spruce)
you don't have:
1- The number of years it will take for the plants to get to the same height(lets say Years = Y)

here is the formula: 4Y + 14 = 6Y + 8

Notice that getting Y(years) correctly by solving the equation will give you the answer, and that the left side would equal the right side (height of hemlock would equal height of the spurce in Y years)

lets solve it out:

4Y + 14 = 6Y + 8
4Y - 6Y = 8 - 14 (switching)
-2Y = -6 (subtracting)
-2 / -2Y = -6 / -2 (getting rid of -2 before the Y and making sure Y is not negative)

Y = 3 (the answer you have)

to get the height of the two plants you subtitute Y in the equation:

4Y + 14 = ? (hemlock)
4(3) + 14 = ?
12 + 14 = 26 inches


6Y + 8 = ? (spurce)
6(3) + 8 = ?
18 + 8 = 26 inches

simply that's it!

:wink:
 

What are linear systems?

A linear system is a set of equations that involve two or more variables and can be solved simultaneously to find the values of those variables that satisfy all the equations.

What is the process for solving linear systems?

The process for solving linear systems involves identifying the variables, writing out the equations, and then using algebraic techniques such as substitution or elimination to find the values of the variables that satisfy all the equations.

What is the significance of "14" Hemlock & 8" Blue Spruce in this linear system?

In this linear system, "14" Hemlock and "8" Blue Spruce represent the number of trees of each type. These values are the variables in the system and solving the system will give the specific values for each type of tree.

Why is it important to solve linear systems?

Solving linear systems is important because it allows us to find the values of multiple variables that satisfy a set of equations. This can be useful in many real-world applications, such as determining the optimal solution to a problem or predicting future outcomes.

What are some techniques for solving linear systems?

Some common techniques for solving linear systems include substitution, elimination, and graphing. Other techniques such as Cramer's rule, Gaussian elimination, and matrix methods can also be used depending on the complexity of the system.

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