Question

An ice skating rink has two ways you can use the facility. The first option is to pay $10 per visit. The second option is to pay $32 per month, and $6 per visit. Write and solve an equation to show the number of visits it would take for the cost of the two plans to be equal.(1 point)
Responses

10x=32+6x ; x=4
10 x equals 32 plus 6 x ; x equals 4

10x=32+6x ; x=8
10 x equals 32 plus 6 x ; x equals 8

10x=32−6x ; x=2
10 x equals 32 minus 6 x ; x equals 2

10=32+6x ; x=−3
10 equals 32 plus 6 x ; x equals negative 3

Answer 1

Solve for x to determine if the equation has one, none, or infinite solutions.
5(2x+2)−4=2x+6 (1 point)
Responses

The equation has one solution: x=0.
The equation has one solution: x equals 0 .

The equation has no solution.
The equation has no solution.

The equation has infinite solutions.
The equation has infinite solutions.

The equation has one solution: x=1.
The equation has one solution: x equals 1 .

Answer 2

To find the number of visits at which the costs of the two options are equal, we can set up an equation based on the costs of each option.

Let \( x \) represent the number of visits.

1. The cost for the first option (paying $10 per visit) is given by:
\[
\text{Cost}_1 = 10x
\]

2. The cost for the second option (paying $32 per month plus $6 per visit) is:
\[
\text{Cost}_2 = 32 + 6x
\]

To find when these two costs are equal, we set the equations equal to each other:
\[
10x = 32 + 6x
\]

Now, let's solve for \( x \):
1. Subtract \( 6x \) from both sides:
\[
10x - 6x = 32
\]
\[
4x = 32
\]

2. Divide both sides by 4:
\[
x = \frac{32}{4} = 8
\]

So, it would take **8 visits** for the costs of the two plans to be equal.

The correct response is:
**10x=32+6x ; x=8**

Answer 3

To determine if the equation \(5(2x + 2) - 4 = 2x + 6\) has one, none, or infinite solutions, we need to solve for \(x\).

Let's start by simplifying the equation step by step.

1. Distribute the 5 on the left side:
\[
5(2x) + 5(2) - 4 = 2x + 6
\]
Which simplifies to:
\[
10x + 10 - 4 = 2x + 6
\]
So,
\[
10x + 6 = 2x + 6
\]

2. Next, subtract \(2x\) from both sides:
\[
10x - 2x + 6 = 6
\]
This simplifies to:
\[
8x + 6 = 6
\]

3. Now, subtract 6 from both sides:
\[
8x = 6 - 6
\]
Which gives:
\[
8x = 0
\]

4. Finally, divide both sides by 8:
\[
x = 0
\]

Since we found a specific solution for \(x\), the equation has **one solution: \(x = 0\)**.

So the correct response is:
**The equation has one solution: x=0.**