The table shows a representation of the number of miles a car drives over time.
| Hours, $x$ | Miles, $y$ |
|---|---|
| 3 | 195 |
| 4 | 260 |
| 5 | 325 |
| 6 | 390 |
Pattern: Each $x$ value is multiplied by 65 to get each $y$ value.
What is the equation for this situation?
Answer (2)
Calculate the slope m using two points from the table: m = 4 − 3 260 − 195 = 65 .
Calculate the y-intercept b by substituting a point and the slope into y = m x + b : 195 = 65 × 3 + b , which gives b = 0 .
Write the equation of the line: y = 65 x .
The equation representing the situation is y = 65 x .
Explanation
Understanding the Problem We are given a table that shows the relationship between the number of hours a car drives ( x ) and the number of miles it travels ( y ). We need to find the equation that represents this relationship.
Calculating the Slope First, let's find the slope of the line. We can use any two points from the table. Let's use the points (3, 195) and (4, 260). The slope, m , is calculated as: m = x 2 − x 1 y 2 − y 1 = 4 − 3 260 − 195 = 1 65 = 65 So, the slope is 65.
Finding the Y-Intercept Now, let's find the y-intercept, b . We can use the slope-intercept form of a linear equation, which is y = m x + b . We can plug in one of the points from the table and the slope we just calculated. Let's use the point (3, 195): 195 = 65 × 3 + b 195 = 195 + b b = 195 − 195 = 0 So, the y-intercept is 0.
Writing the Equation Now we can write the equation of the line using the slope and y-intercept: y = m x + b . Since m = 65 and b = 0 , the equation is: y = 65 x This equation represents the relationship between the number of hours ( x ) and the number of miles ( y ).
Final Answer The equation for this situation is y = 65 x . This means that for every hour the car drives, it travels 65 miles.
Examples
Understanding linear relationships like the one in this problem is useful in many real-life situations. For example, if you are planning a road trip, you can use this equation to estimate how far you will travel in a certain amount of time, assuming you maintain a constant speed. Similarly, businesses can use linear equations to model costs, revenue, and profits based on production levels or sales volumes. This helps in making informed decisions about pricing, production, and resource allocation.
The equation representing the relationship between hours driven and miles traveled is y = 65 x . This indicates that the car travels 65 miles for every hour it drives. The slope of 65 and a y-intercept of 0 verify the linear relationship.
;
