Question

a tree that is 8 feet tall is growing at a rate of 1 foot each year. A different that is 10 feet tall is growing at a rate of 1/2 foot each year.

Write expressions to show how tall each tree will be in x years.

Use your expressions to find how many years it will take for the trees to be the same height.

Answer 1

Let's use x = years and y = height

Start with the 8-foot tree.
It is 8 ft tall now.

For time = x = 0, which is now,
y = 8

For each year that passes, it grows 1 ft.

In 1 year, it grows 1 ft.
In 1 year, it is y = 8 + 1 ft tall

In 2 years, it grows 2 ft.
In 2 years, it is y = 8 + 2 ft tall

In 3 years, it grows 3 ft.
In 3 years, it is 8 + 3 ft tall

In x years, it grows x ft.
In x years, it is y = 8 + x ft tall

Now we have an equation for the first tree:
y = x + 8

Do you understand it so far?

Answer 2

Yes.

Answer 3

Great.
Now we can do a similar thing for the other tree.
It starts at 10 ft tall.

In year zero, when x = 0, y = 10. The tree is 10 ft tall now.

The second tree grows only 0.5 ft per year.

After 1 years, the tree is y = 10 + 0.5 ft tall
After 2 years, the tree is y = 10 + (0.5)(2) = 10 + 1 ft tall
After 3 years, the tree is y = 10 + (0.5)(3) = 10 + 1.5 ft tall
After x years, the tree is y = 10 + (0.5)x = 0.5x + 10 ft tall

The expression for the growth of the second tree is

y = 0.5x + 10

Answer 4

Do you understand how we arrive at the second equation?

Answer 5

Yep.

Answer 6

Thank you, by the way.

Answer 7

We have answered the first question.
The second question is asking when the trees will have the same height.

First tree height: y = x + 8
Second tree height: y = 0.5x + 10

You want the heights to be the same.
You want the height of the first tree, x + 8,
to equal the height of the second tree, 0.5x + 10

Set the two expressions in x equal to each other.

Then solve for x, to find in which year, x, the trees will have the same height.

Answer 8

You're welcome.

Answer 9

What equation do you get?

Answer 10

Year 2?

Answer 11

I didn't get that.
Can you show me the equation you came up with?

Answer 12

Wait, I was wrong.

Answer 13

year one is 10.5 for tree a and 9 for tree b. year 2 is 11 for tree a and 10 for tree b. year 3 is 11.5 for tree a and 11 for tree b. Year 4 is 12 for tree a and 12 for tree b

Answer 14

so year 4?

Answer 15

I think I got tree a and b mixed up but oh well.

Answer 16

You are correct. The answer is year 4.
Yes, you can do it that way, but you are not following the directions.
The problem specifically asks you to use the expressions to find the answer.

This is how the problem wants you to answer the question:

Set the two expressions of the heights of the trees equal, and solve the equation for x.

x + 8 = 0.5x + 10

Subtract 0.5x from both sides:

0.5x + 8 = 10

Subtract 8 from both sides:

0.5x = 2

Divide both sides by 0.5:

x = 4

Answer: The trees will have the same height is year 4.

Answer 17

Oh. Alright. Thanks. :)

Answer 18

You're welcome.
BTW, your method was good, and it worked since you got the correct answer.
The problem is that if the problem wants you to solve it by a specific method, you may not get credit if you solve it a different way.