Determine any data values that are missing from the table, assuming that the data represent a linear function.
| x | y |
| -- | -- |
| 4 | 17 |
| 6 | 19 |
| | 21 |
| 10 | |
a. Missing x: 8
Missing y: 23
b. Missing x: 10
Missing y: 21
c. Missing x: 8
Missing y: 22
d. Missing x: 9
Missing y: 20
Please select the best answer from the choices provided
Answer (2)
Calculate the slope using the given points: m = 6 − 4 19 − 17 = 1 .
Find the equation of the line using point-slope form: y − 17 = 1 ( x − 4 ) , which simplifies to y = x + 13 .
Substitute y = 21 to find the missing x : 21 = x + 13 , so x = 8 .
Substitute x = 10 to find the missing y : y = 10 + 13 , so y = 23 . The final answer is M i ss in g x : 8 an d M i ss in g y : 23 .
Explanation
Understanding the Problem We are given a table with missing values for a linear function and asked to determine those missing values. The given data points are (4, 17) and (6, 19). We need to find the missing x value when y = 21 and the missing y value when x = 10 .
Calculating the Slope First, we need to find the slope of the linear function. The slope, m , is calculated as the change in y divided by the change in x . Using the points (4, 17) and (6, 19), we have: m = x 2 − x 1 y 2 − y 1 = 6 − 4 19 − 17 = 2 2 = 1
Finding the Equation of the Line Now that we have the slope, we can use the point-slope form of a linear equation to find the equation of the line. The point-slope form is given by y − y 1 = m ( x − x 1 ) . Using the point (4, 17) and the slope m = 1 , we get: y − 17 = 1 ( x − 4 ) y = x − 4 + 17 y = x + 13
Finding the Missing x Value Next, we substitute y = 21 into the equation to find the missing x value: 21 = x + 13 x = 21 − 13 x = 8
Finding the Missing y Value Now, we substitute x = 10 into the equation to find the missing y value: y = 10 + 13 y = 23
Final Answer Therefore, the missing x value is 8 and the missing y value is 23. This corresponds to option a.
Examples
Linear functions are incredibly useful in everyday life. For example, if you are saving money at a constant rate, the relationship between the amount of money you save and the time you spend saving can be modeled by a linear function. Similarly, if you are traveling at a constant speed, the relationship between the distance you travel and the time you spend traveling is also a linear function. Understanding linear functions allows you to make predictions and solve problems in these types of situations.
The missing values for the table are x = 8 and y = 23, which corresponds to option a. We determined this by calculating the slope and finding the linear equation. Then we substituted the known values to find the missing ones.
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