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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

3 answers

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

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 .

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.