Question

In a science experiment, the relationship between two variables is given by the equation 3(x−2) = 3x−
. Fill in the blank to create an equation with infinitely many solutions.
2. A company’s profit ( P ) is related to its sales ( s ) by the equation 2P + 4 = 2(P + 2) . Which of the following
statements is true about this equation?
A. The equation has
one unique solution.
C. The equation has
infinitely many
solutions.
B. The equation has
no solution.
D. More information is
needed to determine
the number of
solutions.
3. A charity organization is planning a fundraising event. They have two pricing options: Option 1: $20 per ticket
+ $5 handling fee Option 2: $25 per ticket (no handling fee)
Write a linear equation to represent when these two options would result in the same total cost. Then, determine
if this equation has one solution, infinitely many solutions, or no solution. Explain your reasoning and show your
work

Answer 1

1. Fill in the blank: 3(x − 2) = 3x − __
- Expand left: 3(x − 2) = 3x − 6.
- So fill in 6. The equation 3(x − 2) = 3x − 6 simplifies to 3x − 6 = 3x − 6, which is true for all x → infinitely many solutions.

2. 2P + 4 = 2(P + 2)
- Expand right: 2(P + 2) = 2P + 4, so the equation becomes 2P + 4 = 2P + 4.
- This is an identity true for every P, so the correct choice is C: the equation has infinitely many solutions.

3. Ticket pricing options
- Let t = number of tickets.
- Option 1 cost: 20t + 5. Option 2 cost: 25t.
- Set equal: 20t + 5 = 25t. Solve:
5 = 25t − 20t = 5t → t = 1.
- There is one solution (t = 1). Checking: both costs = $25 when t = 1. Thus the equation has one unique solution.