Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.

Question

Samuel and Josie met for dinner in downtown Dayton. Samuel did flat-rate valet parking for $17. Josie went to another valet and paid $7 up front and $1 for every hour, including the first hour. Ultimately, the friends ended up paying the same amount. How much did each one pay? How long did they stay?

Samuel and Josie both ended up paying $ ___ for ___ hours of parking.

Answer 1

Let's denote the amount of time they parked (in hours) by \( x \).

For Samuel's parking: Samuel paid a flat rate of $17.

For Josie's parking: Josie paid $7 up front and $1 for every hour of parking. Therefore, the total amount paid by Josie can be represented as: \[ \text{Josie's total cost} = 7 + 1 \cdot x = 7 + x \]

Since both Samuel and Josie ended up paying the same amount, we can set up the following equation: \[ 17 = 7 + x \]

Next, we'll solve for \( x \):

1. Subtract 7 from both sides: \[ 17 - 7 = x \] \[ 10 = x \]

So, \( x = 10 \) hours.

Now we can find out how much each paid:

• Samuel's cost: $17
• Josie's cost: \[ 7 + x = 7 + 10 = 17 \]

Thus, both Samuel and Josie paid $17.

Putting it all together: Samuel and Josie both ended up paying $ 17 for 10 hours of parking.

Answer 2

John, a caterer, is investing some money in equipment and employees to help grow his business. Recently he spent $46 on equipment and hired a server who makes $15 per hour. John is hoping to make up these expense at the next job that is scheduled, which pays a base of $47 in addition to $14 per hour that the server works. In theory, this event could pay enough to cancel out John's expenditures. How long would the job have to be? How much would the job pay?

If the job lasted ___ hours, the expenditures and pay would both be $___.

Answer 3

Let's denote the number of hours the server works at the job as \( x \).

John's expenditures:

• He spent $46 on equipment.
• The server costs $15 per hour, which for \( x \) hours would be \( 15x \).

Thus, John's total expenditure can be expressed as: \[ \text{Total Expenditure} = 46 + 15x \]

John's income from the job:

• The job has a base pay of $47.
• Additionally, the server earns $14 per hour worked, which for \( x \) hours would be \( 14x \).

Thus, John's total income from the job can be expressed as: \[ \text{Total Income} = 47 + 14x \]

Setting the expenditures equal to the income: To find out how long the job would have to last for the income to cancel out the expenditures, we set the two equations equal to each other: \[ 46 + 15x = 47 + 14x \]

Now we will solve for \( x \):

1. Subtract \( 14x \) from both sides: \[ 46 + 15x - 14x = 47 \] \[ 46 + x = 47 \]
2. Subtract 46 from both sides: \[ x = 47 - 46 \] \[ x = 1 \]

Thus, the job would have to last 1 hour.

Calculating how much the job would pay: To find out how much the job would pay for 1 hour: \[ \text{Total Income} = 47 + 14(1) = 47 + 14 = 61 \]

So, if the job lasted 1 hour, the expenditures and pay would both be $61.

Putting it all together: If the job lasted 1 hours, the expenditures and pay would both be $61.

Answer 4

Myra wants to buy a charm bracelet. Greenville Fine Jewelry charges $23 per charm, plus $72 for the bracelet. Simon Jewelers, in contrast, charges $24 per charm and $66 for the bracelet. If Myra wants to add a certain number of charms to her bracelet, the cost will be the same at either jewelry shop. How many charms would that be? What would the total cost of the bracelet be?

Answer 5

Let the number of charms Myra wants to add be denoted as \( x \).

Cost at Greenville Fine Jewelry: The total cost for the bracelet and charms can be expressed as: \[ \text{Cost}_{\text{Greenville}} = 72 + 23x \]

Cost at Simon Jewelers: The total cost for the bracelet and charms can be expressed as: \[ \text{Cost}_{\text{Simon}} = 66 + 24x \]

To find the number of charms at which the costs are the same, we set the two equations equal to each other: \[ 72 + 23x = 66 + 24x \]

Now, let's solve for \( x \):

1. Subtract \( 23x \) from both sides: \[ 72 = 66 + 24x - 23x \] \[ 72 = 66 + x \]
2. Subtract 66 from both sides: \[ 72 - 66 = x \] \[ x = 6 \]

So, Myra would need to add 6 charms for the total costs to be the same.

Calculating the total cost of the bracelet with charms: Using either equation to calculate the total cost when \( x = 6 \):

Using Greenville Fine Jewelry: \[ \text{Cost} = 72 + 23(6) = 72 + 138 = 210 \]

Using Simon Jewelers: \[ \text{Cost} = 66 + 24(6) = 66 + 144 = 210 \]

In both cases, the total cost is $210.

Putting it all together: Myra would need to add 6 charms, and the total cost of the bracelet would be $210.

Answer 6

Mrs. Norton, the P.E. teacher, is pairing off students to race against each other. Kinsley can run 6 yards per second, and Francesca can run 9 yards per second. Mrs. Norton decides to give Kinsley a head start of 24 yards since she runs more slowly. Once the students start running, Francesca will quickly catch up to Kinsley. How long will that take? How far will Francesca have to run?

Answer 7

To determine how long it will take for Francesca to catch up with Kinsley, we first establish the distances they will run based on their speeds and the head start given to Kinsley.

Let \( t \) be the time in seconds that Francesca runs.

Kinsley's distance:

• Kinsley receives a 24-yard head start and runs at 6 yards per second, so the distance she covers after \( t \) seconds is: \[ \text{Distance}_{\text{Kinsley}} = 24 + 6t \]

Francesca's distance:

• Francesca runs at 9 yards per second, so the distance she covers after \( t \) seconds is: \[ \text{Distance}_{\text{Francesca}} = 9t \]

Setting the distances equal to find when Francesca catches up: To find when Francesca catches up to Kinsley, we set their distances equal: \[ 24 + 6t = 9t \]

Now, let's solve for \( t \):

1. Rearrange the equation: \[ 24 = 9t - 6t \] \[ 24 = 3t \]
2. Divide both sides by 3: \[ t = \frac{24}{3} = 8 \]

So, it will take 8 seconds for Francesca to catch up to Kinsley.

Calculating the distance Francesca runs: To find out how far Francesca has to run, we can plug \( t = 8 \) seconds into the equation for Francesca's distance: \[ \text{Distance}_{\text{Francesca}} = 9t = 9(8) = 72 \text{ yards} \]

Putting it all together: It will take 8 seconds for Francesca to catch up, and she will have to run 72 yards.

Answer 8

A company that teaches self-improvement seminars is holding one of its seminars in Dover. The company pays a flat fee of $618 to rent a facility in which to hold each session. Additionally, for every attendee who registers, the company must spend $12 to purchase books and supplies. Each attendee will pay $13 for the seminar. Once a certain number of attendee register, the company will be breaking even. How many attendees will that take? What will be the company's total expenses and revenues?

Answer 9

To find the number of attendees needed for the company to break even, we can set up equations for the total expenses and total revenue.

Let's denote the number of attendees as \( x \).

Total Expenses

The total expenses consist of the flat fee for renting the facility and the variable cost for books and supplies per attendee:

• Flat fee for renting: \( 618 \)
• Cost per attendee for books and supplies: \( 12x \)

Thus, the total expenses \( E \) can be expressed as: \[ E = 618 + 12x \]

Total Revenue

The revenue comes from the fees paid by the attendees:

• Revenue per attendee: \( 13 \)

Thus, the total revenue \( R \) can be expressed as: \[ R = 13x \]

Break Even Point

To find the break-even point, we set the total expenses equal to the total revenue: \[ 618 + 12x = 13x \]

Now, let's solve for \( x \):

1. Rearranging gives: \[ 618 = 13x - 12x \] \[ 618 = x \]

Thus, the company needs 618 attendees to break even.

Total Expenses and Revenues at Break Even

To find the total expenses and total revenues when there are 618 attendees, we can substitute \( x = 618 \) back into both equations.

Total Expenses: \[ E = 618 + 12(618) = 618 + 7416 = 8034 \]

Total Revenue: \[ R = 13(618) = 8034 \]

So, at the break-even point:

• The number of attendees is 618.
• The total expenses are $8034.
• The total revenues are $8034.

Answer 10

Use this prompt instead.

Once ___attendees have registered, the company's expenses and receipts will both total $___.

Answer 11

Once 618 attendees have registered, the company's expenses and receipts will both total $8034.