The equation y=2x-5 represents a linear relationship between the miles Alice walks (x) and the time spent walking (y), with a slope of 2 and a y-intercept of -5, where each mile walked increases the time by 2 hours and she starts with a 5-hour deficit.
The equation y=2x−5 illustrates the linear relationship between the distance Alice walks and the time she spends walking. In this context, x represents the distance walked in miles, which is the independent variable, and y represents the corresponding time spent, which is the dependent variable. The equation is in the slope-intercept form y=mx+b, where m is the slope and b is the y-intercept.
The slope of the line (m=2) indicates that for each additional mile Alice walks, her total walking time increases by 2 hours. The y-intercept (b=−5) suggests that Alice starts with a 5-hour deficit, meaning before she even begins walking, she has a time penalty of 5 hours. This could represent the time she spends preparing or the time she initially planned to start earlier. In real-world scenarios, we often see time as the independent variable; however, in this case, distance is taken as an independent variable which is a valid approach depending on what variable is being manipulated or measured with respect to the other.
Using similar equations, such as y = 55x + 75, where y represents the total charge and x is the number of hours worked on a car, we can deduce that the slope of 55 shows the cost per hour and the intercept of 75 indicates a fixed charge independent of the hours worked. The equations provided for displacement and velocity further emphasize the concept of linear relationships in various contexts, particularly within the field of physics, where displacement is often a linear function of time or velocity.
The probable question may be:
Imagine a group of friends, Alice, Bob, and Carol, who enjoy going on nature walks. They decided to record the distance they walked each day. After collecting data for several days, they noticed that the distance covered by Alice in a day (represented by x miles) is related to the time she spends walking, and this can be expressed by the equation y=2x−5.
Additional Information:
Alice, the nature enthusiast, measures her walks in miles (x) and records the corresponding time spent (y). The equation y=2x−5 represents the relationship they discovered. For every additional mile Alice walks, her total time increases by 2 hours, but she always starts 5 hours behind schedule.