Select the approximate value for the solution to the equation in the table below.
| Successive Approximation Intersection Values | |
| :-------------------------------------------------------------: | :----: |
| 33/16 | 17/8 |
| 35/16 | 9/4 |
| 37/16 | 19/8 |
Answer (2)
Calculate the decimal values of the x and y values.
Find the difference between the x and y values for each row.
Since the differences are the same, any of the x or y values can be selected as the approximate solution.
The approximate value for the solution to the equation is 16 35 .
Explanation
Understanding the Problem We are given a table of successive approximation intersection values and asked to select the approximate value for the solution to the equation. The table provides x and y values that are getting closer to the solution. The solution should be where the x and y values are close to each other.
Calculating Decimal Values The x-values are 16 33 = 2.0625 , 16 35 = 2.1875 , and 16 37 = 2.3125 . The corresponding y-values are 8 17 = 2.125 , 4 9 = 2.25 , and 8 19 = 2.375 .
Finding the Differences We calculate the difference between the x and y values for each row: Row 1: ∣2.0625 − 2.125∣ = 0.0625 Row 2: ∣2.1875 − 2.25∣ = 0.0625 Row 3: ∣2.3125 − 2.375∣ = 0.0625 Since the differences are the same, we can take the average of the x and y values for each row and see which average is closest to both x and y values. However, since the differences are all the same, any of the x or y values can be selected as the approximate solution.
Approximating the Solution Since the differences between the x and y values are the same for all rows, we can choose any of the x or y values as an approximate solution. Let's choose the x-values. The x-values are 2.0625, 2.1875, and 2.3125. We can take the average of these values to get a more accurate approximation: 3 2.0625 + 2.1875 + 2.3125 = 3 6.5625 = 2.1875 . This is equal to 16 35 .
Final Answer The approximate value for the solution to the equation is 16 35 or 2.1875 .
Examples
In numerical analysis, successive approximation is a method of finding better and better approximate values to the solution of an equation. This method is used in many real-world applications, such as finding the roots of a function or solving a system of equations. For example, engineers use successive approximation to design bridges and buildings, and economists use it to model the behavior of financial markets. By iteratively refining the approximation, we can get closer and closer to the true solution.
The approximate value for the solution to the equation is 16 35 or 2.1875 . This value was determined by calculating and comparing the differences between x and y values in the provided table. As all differences were equal, we could select any corresponding values as the solution.
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