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

Elizabeth loves riding Ferris wheels and roller coasters. While visiting the Washington County Fair, she first went on the Ferris wheel 5 times and the roller coaster 4 times, using a total of 34 tickets. Then, after taking a break and having a snack, Elizabeth went on the Ferris wheel 2 times and the roller coaster 2 times, using a total of 16 tickets. How many tickets does it take to ride each attraction?

It takes ___ tickets to ride the Ferris wheel, and ___ tickets to ride the roller coaster.

11 answers

Let's define the variables:

  • Let \( F \) be the number of tickets for a Ferris wheel ride.
  • Let \( R \) be the number of tickets for a roller coaster ride.

From the information given, we can set up the following system of equations based on Elizabeth's rides:

  1. For the first scenario (5 rides on the Ferris wheel and 4 rides on the roller coaster totaling 34 tickets): \[ 5F + 4R = 34 \quad \text{(Equation 1)} \]

  2. For the second scenario (2 rides on the Ferris wheel and 2 rides on the roller coaster totaling 16 tickets): \[ 2F + 2R = 16 \quad \text{(Equation 2)} \]

Now we can simplify Equation 2 by dividing everything by 2: \[ F + R = 8 \quad \text{(Equation 2 simplified)} \]

Now, we will use the elimination method to solve the system of equations. We have:

  1. \( 5F + 4R = 34 \)
  2. \( F + R = 8 \)

We can solve Equation 2 for \( R \): \[ R = 8 - F \]

Now, substitute this expression for \( R \) into Equation 1: \[ 5F + 4(8 - F) = 34 \]

Now simplify and solve for \( F \): \[ 5F + 32 - 4F = 34 \] \[ F + 32 = 34 \] \[ F = 34 - 32 \] \[ F = 2 \]

Now that we have \( F \), we can find \( R \) using Equation 2: \[ F + R = 8 \] \[ 2 + R = 8 \] \[ R = 8 - 2 \] \[ R = 6 \]

Thus, it takes 2 tickets to ride the Ferris wheel and 6 tickets to ride the roller coaster.

So, filling in the blanks: It takes 2 tickets to ride the Ferris wheel, and 6 tickets to ride the roller coaster.

Kari likes to knit hats and mittens for friends and family. Last fall, she knitted 1 hat and 1 pair of mittens, which took a total of 16 hours. This fall, she knitted 1 hat and 5 pairs of mittens, which took a total of 60 hours. If each hat takes the same amount of time and each pair of mittens takes the same time, how long does it take Kari to knit each item?

It takes Kari
hours to knit a hat and
hours to knit a pair of mittens.

Let's define the variables:

  • Let \( H \) be the number of hours it takes to knit a hat.
  • Let \( M \) be the number of hours it takes to knit a pair of mittens.

From the information given, we can set up the following system of equations based on Kari's knitting:

  1. For the first scenario (1 hat and 1 pair of mittens taking a total of 16 hours): \[ H + M = 16 \quad \text{(Equation 1)} \]

  2. For the second scenario (1 hat and 5 pairs of mittens taking a total of 60 hours): \[ H + 5M = 60 \quad \text{(Equation 2)} \]

Now, we can use the elimination method to solve the system of equations. We have:

  1. \( H + M = 16 \)
  2. \( H + 5M = 60 \)

First, we will solve Equation 1 for \( H \): \[ H = 16 - M \]

Now, we substitute this expression for \( H \) in Equation 2: \[ (16 - M) + 5M = 60 \]

Now simplify and solve for \( M \): \[ 16 - M + 5M = 60 \] \[ 16 + 4M = 60 \] \[ 4M = 60 - 16 \] \[ 4M = 44 \] \[ M = \frac{44}{4} \] \[ M = 11 \]

Now that we have \( M \), we can find \( H \) using Equation 1: \[ H + M = 16 \] \[ H + 11 = 16 \] \[ H = 16 - 11 \] \[ H = 5 \]

Thus, it takes Kari 5 hours to knit a hat and 11 hours to knit a pair of mittens.

So, filling in the blanks: It takes Kari 5 hours to knit a hat and 11 hours to knit a pair of mittens.

A coffee shop is having a sale on prepackaged coffee and tea. Yesterday they sold 26 packages of coffee and 28 packages of tea, for which customers paid a total of $322. The day before, 19 packages of coffee and 24 packages of tea was sold, which brought in a total of $253. How much does each package cost?

Per package, coffee costs $
and tea costs $
.

Let's define the variables:

  • Let \( C \) be the price of a package of coffee.
  • Let \( T \) be the price of a package of tea.

We can set up the following system of equations based on the sales information provided:

  1. For the first day (26 packages of coffee and 28 packages of tea totaling $322): \[ 26C + 28T = 322 \quad \text{(Equation 1)} \]

  2. For the second day (19 packages of coffee and 24 packages of tea totaling $253): \[ 19C + 24T = 253 \quad \text{(Equation 2)} \]

Now we can solve this system of equations using elimination or substitution. We'll use elimination for this example.

First, let's multiply Equation 1 and Equation 2 to align the coefficients of \( C \). To eliminate \( C \), we will multiply Equation 1 by \( 19 \) and Equation 2 by \( 26 \):

  • Multiply Equation 1 by \( 19 \): \[ 19(26C) + 19(28T) = 19(322) \] \[ 494C + 532T = 6118 \quad \text{(Equation 3)} \]

  • Multiply Equation 2 by \( 26 \): \[ 26(19C) + 26(24T) = 26(253) \] \[ 494C + 624T = 6578 \quad \text{(Equation 4)} \]

Now, we'll subtract Equation 3 from Equation 4 to eliminate \( C \): \[ (494C + 624T) - (494C + 532T) = 6578 - 6118 \] \[ 624T - 532T = 460 \] \[ 92T = 460 \] \[ T = \frac{460}{92} = 5 \]

Now that we know \( T = 5 \), we can substitute this value back into one of the original equations to solve for \( C \). We'll use Equation 1: \[ 26C + 28(5) = 322 \] \[ 26C + 140 = 322 \] \[ 26C = 322 - 140 \] \[ 26C = 182 \] \[ C = \frac{182}{26} = 7 \]

Thus, the cost per package is:

  • Coffee costs $7
  • Tea costs $5

So, filling in the blanks: Per package, coffee costs $7 and tea costs $5.

A local service organization is wrapping gifts at the mall to raise money for charity. Yesterday, they wrapped 27 small gifts and 38 large gifts, earning a total of $412. Today, they wrapped 27 small gifts and 16 large gifts, and earned $236. How much did they charge to wrap the gifts?

The organization charges $
to wrap a small gift and $
to wrap a large one.

Let's define the variables:

  • Let \( S \) be the price to wrap a small gift.
  • Let \( L \) be the price to wrap a large gift.

From the information given, we can set up the following system of equations based on the amounts earned:

  1. For yesterday (27 small gifts and 38 large gifts totaling $412): \[ 27S + 38L = 412 \quad \text{(Equation 1)} \]

  2. For today (27 small gifts and 16 large gifts totaling $236): \[ 27S + 16L = 236 \quad \text{(Equation 2)} \]

Now we can solve this system of equations using elimination.

First, we'll subtract Equation 2 from Equation 1 to eliminate \( S \): \[ (27S + 38L) - (27S + 16L) = 412 - 236 \] This simplifies to: \[ 38L - 16L = 176 \] \[ 22L = 176 \] \[ L = \frac{176}{22} = 8 \]

Now that we know \( L = 8 \), we can substitute this value back into either of the original equations to solve for \( S \). We'll use Equation 2: \[ 27S + 16(8) = 236 \] \[ 27S + 128 = 236 \] \[ 27S = 236 - 128 \] \[ 27S = 108 \] \[ S = \frac{108}{27} = 4 \]

Thus, the organization charges:

  • $4 to wrap a small gift
  • $8 to wrap a large gift

So, filling in the blanks: The organization charges $4 to wrap a small gift and $8 to wrap a large one.

Brett received some gift cards for music and movie downloads for his birthday. Using one of them, he downloaded 14 songs and 17 movies, which cost a total of $283. Using another, he purchased 4 songs and 17 movies, which cost a total of $263. How much does each download cost?

Downloads cost $
for a song and $
for a movie.

Let's define the variables:

  • Let \( S \) be the cost to download a song.
  • Let \( M \) be the cost to download a movie.

From the information given, we can set up the following system of equations based on Brett's downloads:

Equation 1: For the first purchase (14 songs and 17 movies totaling $283): \[ 14S + 17M = 283 \quad \text{(Equation 1)} \]

Equation 2: For the second purchase (4 songs and 17 movies totaling $263): \[ 4S + 17M = 263 \quad \text{(Equation 2)} \]

Next, we can solve this system of equations. We will use elimination to eliminate \( M \).

First, we'll subtract Equation 2 from Equation 1 to eliminate \( M \):

\[ (14S + 17M) - (4S + 17M) = 283 - 263 \]

This simplifies to: \[ 14S - 4S = 20 \] \[ 10S = 20 \] \[ S = \frac{20}{10} = 2 \]

Now that we know \( S = 2 \), we can substitute this value back into one of the original equations to solve for \( M \). We'll use Equation 2: \[ 4(2) + 17M = 263 \] \[ 8 + 17M = 263 \] \[ 17M = 263 - 8 \] \[ 17M = 255 \] \[ M = \frac{255}{17} = 15 \]

Thus, the costs for downloads are:

  • $2 for a song
  • $15 for a movie

So, filling in the blanks: Downloads cost $2 for a song and $15 for a movie.

Leah and Molly went to an arcade where the machines took tokens. Leah played 5 games of skee ball and 6 games of pinball, using a total of 22 tokens. At the same time, Molly played 5 games of skee ball and 5 games of pinball, using up 20 tokens. How many tokens does each game require?

Every game of skee ball requires
tokens, and every game of pinball requires
tokens.

Let's define the variables:

  • Let \( S \) be the number of tokens required to play one game of skee ball.
  • Let \( P \) be the number of tokens required to play one game of pinball.

From the information given, we can set up the following system of equations based on Leah's and Molly's token usage:

  1. For Leah (5 games of skee ball and 6 games of pinball, totaling 22 tokens): \[ 5S + 6P = 22 \quad \text{(Equation 1)} \]

  2. For Molly (5 games of skee ball and 5 games of pinball, totaling 20 tokens): \[ 5S + 5P = 20 \quad \text{(Equation 2)} \]

Now, we can simplify Equation 2 by dividing it by 5: \[ S + P = 4 \quad \text{(Equation 2 simplified)} \]

Now, we have the following system:

  1. \( 5S + 6P = 22 \)
  2. \( S + P = 4 \)

Next, let's solve Equation 2 for \( S \): \[ S = 4 - P \]

Now we substitute this expression for \( S \) into Equation 1: \[ 5(4 - P) + 6P = 22 \] Expanding this gives: \[ 20 - 5P + 6P = 22 \] \[ 20 + P = 22 \] \[ P = 22 - 20 \] \[ P = 2 \]

Now that we have \( P \), we can find \( S \) using Equation 2: \[ S + P = 4 \] \[ S + 2 = 4 \] \[ S = 4 - 2 \] \[ S = 2 \]

Thus, each game requires:

  • 2 tokens for a game of skee ball
  • 2 tokens for a game of pinball

So, filling in the blanks: Every game of skee ball requires 2 tokens, and every game of pinball requires 2 tokens.