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At the beginning of the month, a song on a streaming service already had 1,000 listens, and was getting 38 new listens each day. Meanwhile, a different song already had 900 listens, and was getting 40 new listens each day. Write an equation to show the number of days it would take for the two songs to have the same number of listens? Use x as your variable.

11 answers

Let \( x \) represent the number of days that have passed.

The number of listens for the first song after \( x \) days can be represented as: \[ 1000 + 38x \]

The number of listens for the second song after \( x \) days can be represented as: \[ 900 + 40x \]

To find the number of days it would take for the two songs to have the same number of listens, we need to set the two expressions equal to each other: \[ 1000 + 38x = 900 + 40x \]

This is the equation that can be solved for \( x \) to determine the number of days until the listens are equal.

Simplify the answer

Sure! Let's simplify the equation we set up:

Starting with the equation: \[ 1000 + 38x = 900 + 40x \]

First, we will subtract \( 38x \) from both sides: \[ 1000 = 900 + 2x \]

Next, subtract 900 from both sides: \[ 100 = 2x \]

Finally, divide both sides by 2: \[ x = 50 \]

So, it will take 50 days for the two songs to have the same number of listens.

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.

Let \( x \) represent the number of visits to the ice skating rink.

For the first option (paying $10 per visit), the total cost can be represented as: \[ \text{Cost}_1 = 10x \]

For the second option (paying $32 per month and $6 per visit), the total cost can be represented as: \[ \text{Cost}_2 = 32 + 6x \]

To find the number of visits \( x \) where the costs are equal, set the two equations equal to each other: \[ 10x = 32 + 6x \]

Now, let's solve the equation:

  1. Subtract \( 6x \) from both sides: \[ 10x - 6x = 32 \] \[ 4x = 32 \]

  2. Divide both sides by 4: \[ x = 8 \]

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

At the beginning of the month, a song on a streaming service had 700 listens, and it was getting 52 new listens each day. Meanwhile, a different song had 600 listens, and it was getting 56 new listens each day. When will the two songs have the same number of listens?
The songs will have the same number of listens after how many days?

To find out when the two songs will have the same number of listens, let's set up an equation using the information given.

Let \( x \) represent the number of days that have passed.

For the first song, the total number of listens after \( x \) days is: \[ 700 + 52x \]

For the second song, the total number of listens after \( x \) days is: \[ 600 + 56x \]

To find out when the two songs will have the same number of listens, set the two expressions equal to each other: \[ 700 + 52x = 600 + 56x \]

Now, we will solve this equation:

  1. Subtract \( 52x \) from both sides: \[ 700 = 600 + 4x \]

  2. Subtract 600 from both sides: \[ 100 = 4x \]

  3. Divide both sides by 4: \[ x = 25 \]

So, the two songs will have the same number of listens after 25 days.

Solve for x to determine if the equation has one, none, or infinite solutions.
11x = 3 (7x - 1) - 10x

To determine the number of solutions for the equation \( 11x = 3(7x - 1) - 10x \), we will first simplify the right side of the equation and then solve for \( x \).

  1. Start with the original equation: \[ 11x = 3(7x - 1) - 10x \]

  2. Distribute \( 3 \) on the right side: \[ 11x = 21x - 3 - 10x \]

  3. Combine like terms on the right side: \[ 11x = 21x - 10x - 3 \] \[ 11x = 11x - 3 \]

  4. Now subtract \( 11x \) from both sides: \[ 11x - 11x = 11x - 11x - 3 \] \[ 0 = -3 \]

The equation \( 0 = -3 \) is a contradiction, meaning that there are no solutions to the equation \( 11x = 3(7x - 1) - 10x \).

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

To solve the equation \( 5(2x + 2) - 4 = 2x + 6 \) and determine if it has one, none, or infinite solutions, follow these steps:

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

  2. Simplify the left side: \[ 10x + 6 = 2x + 6 \]

  3. Now, subtract \( 2x \) from both sides: \[ 10x - 2x + 6 = 6 \] \[ 8x + 6 = 6 \]

  4. Next, subtract \( 6 \) from both sides: \[ 8x = 0 \]

  5. Divide by \( 8 \) to solve for \( x \): \[ x = 0 \]

Since we found a specific value for \( x \), the equation has one solution, which is \( x = 0 \).