An amusement park has two new rollercoasters. The number of people in line to ride each rollercoaster, P, varies depending on the time, t, in hours since the park opened. The length of the rollercoaster lines for any given time is modeled by the following system of linear equations. [P = 6t + 4 [P= 10t Solve the system of equations graphically. How long after the amusement park opens will there be an equal number of people in each rollercoaster line? How many people are in each line when the two lines are equal? Explain your answer, providing calculations, drawings, or any other strategy that will support your reasoning.
An amusement park has two new rollercoasters. The number of people in line to ride each rollercoaster, P, varies depending on the time, t, in hours since the park opened. The length of the rollercoaster lines for any given time is modeled by the following system of linear equations. [P = 6t + 4 [P= 10t Solve the system of equations graphically. How long after the amusement park opens will there be an equal number of people in each rollercoaster line? How many people are in each line when the two lines are equal? Explain your answer, providing calculations, drawings, or any other strategy that will support your reasoning.
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![**Title: Solving Linear Equations for Rollercoaster Line Predictions**
**Overview:**
An amusement park features two new rollercoasters. The number of people in line for each rollercoaster, denoted as \( P \), varies with time \( t \) (in hours since the park opened). The lines are modeled by the following linear equations:
\[
\begin{align*}
1. \quad & P = 6t + 4 \\
2. \quad & P = 10t
\end{align*}
\]
**Problem Statement:**
Solve the system of equations graphically to determine:
1. How long after the park opens will there be an equal number of people in each rollercoaster line?
2. How many people will be in each line at this time?
**Solution Approach:**
- **Graphing the Equations:**
- **Equation 1**: \( P = 6t + 4 \)
- Y-intercept (when \( t=0 \)): \( P = 4 \)
- Slope: 6 (line rises 6 units for every 1 unit increase in \( t \))
- **Equation 2**: \( P = 10t \)
- Y-intercept (when \( t=0 \)): \( P = 0 \)
- Slope: 10 (line rises 10 units for every 1 unit increase in \( t \))
- **Finding the Intersection:**
By setting the equations equal to each other to find when the lines intersect:
\[
6t + 4 = 10t
\]
\[
4 = 4t
\]
\[
t = 1
\]
After 1 hour, the number of people in each line will be equal.
- **Calculating the Number of People:**
Substitute \( t = 1 \) back into either equation to find \( P \):
Using \( P = 10t \):
\[
P = 10 \times 1 = 10
\]
Therefore, 1 hour after the park opens, there will be 10 people in each line.
**Conclusion:**
After solving graphically, we determine that precisely 1 hour after opening, both rollercoaster lines will have an equal number of](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbc9d167a-3919-4c8c-a43c-665a91017611%2F96013a88-2197-4401-824d-eaf543dd88e4%2Fdwsgjwl_processed.png&w=3840&q=75)
Transcribed Image Text:**Title: Solving Linear Equations for Rollercoaster Line Predictions**
**Overview:**
An amusement park features two new rollercoasters. The number of people in line for each rollercoaster, denoted as \( P \), varies with time \( t \) (in hours since the park opened). The lines are modeled by the following linear equations:
\[
\begin{align*}
1. \quad & P = 6t + 4 \\
2. \quad & P = 10t
\end{align*}
\]
**Problem Statement:**
Solve the system of equations graphically to determine:
1. How long after the park opens will there be an equal number of people in each rollercoaster line?
2. How many people will be in each line at this time?
**Solution Approach:**
- **Graphing the Equations:**
- **Equation 1**: \( P = 6t + 4 \)
- Y-intercept (when \( t=0 \)): \( P = 4 \)
- Slope: 6 (line rises 6 units for every 1 unit increase in \( t \))
- **Equation 2**: \( P = 10t \)
- Y-intercept (when \( t=0 \)): \( P = 0 \)
- Slope: 10 (line rises 10 units for every 1 unit increase in \( t \))
- **Finding the Intersection:**
By setting the equations equal to each other to find when the lines intersect:
\[
6t + 4 = 10t
\]
\[
4 = 4t
\]
\[
t = 1
\]
After 1 hour, the number of people in each line will be equal.
- **Calculating the Number of People:**
Substitute \( t = 1 \) back into either equation to find \( P \):
Using \( P = 10t \):
\[
P = 10 \times 1 = 10
\]
Therefore, 1 hour after the park opens, there will be 10 people in each line.
**Conclusion:**
After solving graphically, we determine that precisely 1 hour after opening, both rollercoaster lines will have an equal number of
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