Use the drawing tool(s) to form the correct answers on the provided answer space.
Beth is tracking how many miles she walks in a week. Currently, she has walked one mile and plans to walk at a brisk pace of five miles per hour for the rest of the week. Her brother Jonathan is also tracking the miles he walks in a week. He has already walked three miles and plans to walk at a slower pace of three and a half miles per hour for the rest of the week.
The system of equations below represents the miles, [tex]y[/tex], walked by Beth and Jonathan in [tex]x[/tex] hours.
[tex]
\begin{array}{l}
y=5 x+1 \
y=3 \frac{1}{2} x+3
\end{array}
[/tex]
Part A: Use the ray tool to graph the system of equations on the coordinate plane.
Part B: Use the point tool to select an approximate amount of time Beth and Jonathan will each have to walk this week in order for them to have walked the same distance.
Answer (2)
To find when Beth and Jonathan have walked the same distance, we set their distance equations equal: 5 x + 1 = 3.5 x + 3 . Solving leads us to find that this occurs after approximately 1.33 hours. This means both will have walked the same distance at this time.
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Set Beth's distance equal to Jonathan's distance: 5 x + 1 = 3.5 x + 3 .
Simplify the equation by subtracting 3.5 x and 1 from both sides: 1.5 x = 2 .
Solve for x by dividing both sides by 1.5 : x = 3 4 .
The time when they have walked the same distance is approximately 1.33 hours: 3 4
Explanation
Understanding the Problem We are given a system of equations representing the distance Beth and Jonathan walk in terms of time. Beth's distance is represented by y = 5 x + 1 , and Jonathan's distance is represented by y = 3.5 x + 3 , where y is the distance in miles and x is the time in hours. We need to find the time x when they have walked the same distance, which means we need to find the intersection point of the two lines.
Setting up the Equation To find the time when Beth and Jonathan have walked the same distance, we need to solve the system of equations:
y = 5 x + 1 y = 3.5 x + 3
Since both equations are equal to y , we can set them equal to each other:
5 x + 1 = 3.5 x + 3
Solving for x Now, we solve for x :
Subtract 3.5 x from both sides:
5 x − 3.5 x + 1 = 3.5 x − 3.5 x + 3 1.5 x + 1 = 3
Subtract 1 from both sides:
1.5 x + 1 − 1 = 3 − 1 1.5 x = 2
Calculating the Time Divide both sides by 1.5:
x = 1.5 2 = 2 3 2 = 1 2 × 3 2 = 3 4
So, x = 3 4 hours, which is approximately 1.33 hours.
Final Answer Therefore, Beth and Jonathan will have walked the same distance after approximately 1.33 hours.
Examples
Imagine you and a friend are saving money. You start with $1 and save $5 per week. Your friend starts with $3 and saves $3.50 per week. This problem helps you determine how many weeks it will take for both of you to have the same amount of money saved. Understanding linear equations and systems of equations can guide you in scenarios such as financial planning, comparing costs, or determining break-even points. This algebraic approach ensures you can make informed decisions based on different rates and initial values.
