Question

A small town has two local high schools. High School A currently has 700 students and is projected to grow by 65 students each year. High School B currently has 900 students and is projected to grow by 40 students each year. Let A(t) represent the number of students in High School A in t years, and let B(t) represent the number of students in High School B after t years. Write the equation for each function and determine how many students would be in each high school in the year they are projected to have the same number of students. A(t) = B(t) =

Answer 1

1. A(t) = 65t + 700

2.B(t) = 40t + 900

3. High School A and High School B will both have 1,220 students in the 8th year

Explanation:

1. The equation for the number of students in High School A represents a linear function.

⭐ What is a linear function?

• A linear function is a type of equation where every y-value increases by a constant, additive amount
• One way to write the equation for a linear function is:
  `y = mx+b`, where m is the constant, additive amount, and b is the y-intercept, or the initial value.

Let's write the equation for High School A in the
`y = mx+b` format, known as slope-intercept form:

• The constant, additive amount for High School A is 65 (m)
• The initial value for High School A is 700 (b)

∴ High School A:
`y = 65x + 700`

2. Let's write the equation for High School B in the
`y = mx + b` format, known as slope-intercept form:

• The constant, additive amount for High School B is 40 (m)
• The initial value for High School B is 900 (b)

∴ High School B:
`y = 40x + 900`

3. To find at what year High School A and High School B will have the same number of students, we need to solve a system of linear equations.

⭐What is a system of linear equations?

• A system of linear equations is two or more linear equations that intersect at one point (x,y)

For this problem, let's set both linear equations equal to each other to see at what point will the high school populations be the same.

`A(t) = B(t)`

`65t + 700 = 40t + 900`

`25t = 200`

`t = 8`

Now we know that in the 8th year, High School A will have the same population as High School B.

We need to find what the population will be in year 8.

Thus, substitute the value of t into one of the functions and solve.

I am choosing to substitute t into A(t), but you can also do B(t).

`A(8) = 65(8) + 700`

`A(8) = 520 + 700`

`A(8) = 1,220`

⚠️!!! CAUTION !!! ⚠️

Some people may stop at this point and write that in the 8th year, both high schools will have a population of 1,220 students.

However, you should also substitute 8 into the other function you didn't substitute it into to make sure that 8 is correct.

`B(8) = 40(8) + 900`

`B(8) = 320 + 900`

`B(8) = 1,220`

∴ In the 8th year, both high schools will have a population of 1,220 students.