What is the closest approximate solution to the system of equations?
$(-5,-7.2)$
$(-5,-6.8)$
$(-5,-6.2)$
$(-5,-5.9)$
Answer (1)
Calculate the midpoint of the y-values: 2 − 7.2 + ( − 5.9 ) = − 6.55 .
Find the absolute differences between each y-value and the midpoint: 0.65, 0.25, 0.35, 0.65.
Identify the smallest difference: 0.25, corresponding to the point ( − 5 , − 6.8 ) .
The closest approximate solution is ( − 5 , − 6.8 ) .
Explanation
Understanding the Problem We are given four points: ( − 5 , − 7.2 ) , ( − 5 , − 6.8 ) , ( − 5 , − 6.2 ) , and ( − 5 , − 5.9 ) . We are asked to find the closest approximate solution to a system of equations, but the system of equations itself is not provided. This means we need to determine which of the given points is the 'best' approximation without knowing the actual solution.
Finding the Midpoint Since we don't have the system of equations, we can't find the exact solution. Instead, let's find the midpoint of the y-values of the given points. This will give us a 'center' point, and we can choose the point closest to this center. The y-values are -7.2, -6.8, -6.2, and -5.9. The midpoint is calculated as follows: 2 − 7.2 + ( − 5.9 ) = 2 − 13.1 = − 6.55 So, the 'center' y-value is -6.55.
Finding the Closest Point Now, we need to find which of the given y-values is closest to -6.55. We can calculate the absolute differences:
∣ − 7.2 − ( − 6.55 ) ∣ = ∣ − 7.2 + 6.55∣ = ∣ − 0.65∣ = 0.65
∣ − 6.8 − ( − 6.55 ) ∣ = ∣ − 6.8 + 6.55∣ = ∣ − 0.25∣ = 0.25
∣ − 6.2 − ( − 6.55 ) ∣ = ∣ − 6.2 + 6.55∣ = ∣0.35∣ = 0.35
∣ − 5.9 − ( − 6.55 ) ∣ = ∣ − 5.9 + 6.55∣ = ∣0.65∣ = 0.65
The smallest difference is 0.25, which corresponds to the point ( − 5 , − 6.8 ) .
Conclusion Therefore, the closest approximate solution to the system of equations among the given points is ( − 5 , − 6.8 ) .
Examples
In GPS navigation, when your device tries to pinpoint your location using signals from multiple satellites, it often gets slightly different readings. The GPS uses a method similar to finding the closest point to reconcile these readings and give you the most accurate location possible. By averaging the coordinates and finding the nearest data point, the GPS provides a reliable estimate, even if the initial signals vary slightly. This ensures that your map directions are as precise as possible, guiding you effectively to your destination.
