2. 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
To find the relationship between x (time) and y (distance), we can plot the data points on a graph.
The points (8, 40), (5, 25), (7, 35), (2, 10), and (9, 45) represent the time and distance pairs.
Plotting these points on a graph, we get:
(8, 40)
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The points roughly form a line, indicating a linear relationship between time (x) and distance (y).
The equation of a line can be represented as y = mx + b, where m is the slope of the line and b is the y-intercept.
To find the slope (m), we can use the formula:
m = (y2 - y1) / (x2 - x1)
Let's use the points (5, 25) and (7, 35) to find the slope:
m = (35 - 25) / (7 - 5)
m = 10 / 2
m = 5
So, the slope of the line is 5.
To find the y-intercept (b), we can use any of the points. Let's use (5, 25):
y = mx + b
25 = 5(5) + b
25 = 25 + b
b = 25 - 25
b = 0
Therefore, the equation that represents the relationship between time (x) and distance (y) is:
y = 5x + 0
y = 5x
To find a relationship between x (time in minutes) and y (distance traveled in meters) based on the given table, we can plot the points on a coordinate graph and see if there is a pattern or trend.
Plotting the points:
(x, y) = (8, 40)
(x, y) = (5, 25)
(x, y) = (7, 35)
(x, y) = (2, 10)
(x, y) = (9, 45)
Now let's connect the points:
(8, 40) --- (5, 25) --- (7, 35) --- (2, 10) --- (9, 45)
By connecting these points, we can observe an increasing trend in the distance traveled as the time increases.
To find the equation that represents this relationship, we can use the slope-intercept form of a linear equation, which is given by y = mx + b, where m is the slope and b is the y-intercept.
Let's calculate the slope (m) using two of the points, say (5, 25) and (8, 40):
m = (y2 - y1) / (x2 - x1)
= (40 - 25) / (8 - 5)
= 15 / 3
= 5
Using the point-slope form of a linear equation with the calculated slope, the equation becomes:
y - y1 = m(x - x1)
y - 25 = 5(x - 5)
Expanding the equation further:
y - 25 = 5x - 25
Finally, let's solve for y:
y = 5x
Therefore, the equation that represents the relationship between x (time in minutes) and y (distance traveled in meters) is y = 5x.
To find the equation that represents the relationship between x (time in minutes) and y (distance traveled in meters), we can use the method of linear regression. This will help us find a line of best fit for the given data.
Step 1: Organize the data in a table:
x y
8 40
5 25
7 35
2 10
9 45
Step 2: Calculate the mean values of x and y:
Mean of x = (8 + 5 + 7 + 2 + 9) / 5 = 31 / 5 = 6.2
Mean of y = (40 + 25 + 35 + 10 + 45) / 5 = 155 / 5 = 31
Step 3: Calculate the deviations from the mean for x and y:
Deviation of x = x - Mean of x
Deviation of y = y - Mean of y
For each data point, calculate the deviations from the mean:
x y Deviation of x Deviation of y
8 40 1.8 9
5 25 -1.2 -6
7 35 0.8 4
2 10 -4.2 -21
9 45 2.8 14
Step 4: Calculate the product of the deviations:
Product of deviations = Deviation of x * Deviation of y
For each data point, calculate the product of deviations:
x y Deviation of x Deviation of y Product of deviations
8 40 1.8 9 16.2
5 25 -1.2 -6 7.2
7 35 0.8 4 3.2
2 10 -4.2 -21 88.2
9 45 2.8 14 39.2
Step 5: Calculate the squared deviations of x:
Squared deviation of x = (Deviation of x)^2
For each data point, calculate the squared deviation of x:
x y Deviation of x Deviation of y Product of deviations Squared deviation of x
8 40 1.8 9 16.2 3.24
5 25 -1.2 -6 7.2 1.44
7 35 0.8 4 3.2 0.64
2 10 -4.2 -21 88.2 17.64
9 45 2.8 14 39.2 7.84
Step 6: Calculate the slope (m) of the line:
m = Sum of (Product of deviations) / Sum of (Squared deviation of x)
m = (16.2 + 7.2 + 3.2 + 88.2 + 39.2) / (3.24 + 1.44 + 0.64 + 17.64 + 7.84)
m = 154 / 30.8
m ≈ 5
Step 7: Calculate the y-intercept (c) of the line:
c = Mean of y - (m * Mean of x)
c = 31 - (5 * 6.2)
c = 31 - 31
c = 0
Step 8: Write the equation in the form of y = mx + c:
y = 5x + 0
Therefore, the equation that represents the relationship between x (time in minutes) and y (distance traveled in meters) is y = 5x.