Given the table for x, time in minutes and y, distance traveled in meters:
x - time (minutes) 8 5 7 2 9
y - distance (meters) 40 25 35 10 45
. Every minute meters are traveled.
b. Write an equation to show the relationship between x and y.
c. The relationship is Response area because the y-intercept is Response area
b. The equation for the relationship between x and y can be written as: y = mx + b, where m is the rate of distance traveled per minute and b is the y-intercept.
c. The relationship is linear because the rate of distance traveled per minute is constant. The y-intercept is the distance traveled when x = 0, which is 0 meters in this case.
To write an equation to show the relationship between x and y, we can use the equation of a line: y = mx + b.
First, let's calculate the rate of change (m) between x and y by taking the change in y divided by the change in x for any two points:
For the points (2, 10) and (9, 45):
m = (45 - 10) / (9 - 2)
m = 35 / 7
m = 5
Now let's find the y-intercept (b). We can choose any point (x, y) and substitute them into the equation:
Using the point (2, 10):
10 = 5(2) + b
10 = 10 + b
10 - 10 = b
b = 0
Combining the slope (m = 5) and y-intercept (b = 0), we can write the equation for the relationship between x and y:
y = 5x + 0
y = 5x
The equation for the relationship between x and y is y = 5x.
The relationship is a direct variation because the y-intercept is 0.
To find the equation that represents the relationship between x (time in minutes) and y (distance traveled in meters), we can use the given data points from the table.
Looking at the data, we can see that as time increases, the distance also increases. This suggests that there is a linear relationship between x and y.
Let's first plot the data points on a coordinate plane, with x-values on the x-axis and y-values on the y-axis:
(x, y)
(8, 40)
(5, 25)
(7, 35)
(2, 10)
(9, 45)
Now, we can find the slope of the line that passes through these points. The slope (m) is given by the formula:
m = (change in y) / (change in x)
Taking two points on the graph, (8, 40) and (5, 25), we can calculate the change in y and change in x:
change in y = 25 - 40 = -15
change in x = 5 - 8 = -3
m = (-15) / (-3) = 5
Now that we have the slope, we can use the point-slope form of a linear equation, which is:
y - y1 = m(x - x1)
Choosing the point (8, 40) as our reference point, we can substitute the values into the equation:
y - 40 = 5(x - 8)
Simplifying the equation, we get:
y - 40 = 5x - 40
Finally, we can rearrange the equation to have y on one side:
y = 5x
So, the equation that represents the relationship between x and y is y = 5x.
Moving on to part c of the question, the y-intercept is the point at which the line crosses the y-axis. In this case, since the equation is y = 5x, the y-intercept is 0. This is because when x is 0, the distance traveled (y) is also 0.
Therefore, the relationship is y = 5x, and the y-intercept is 0.