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Difficulty: MediumIntegers, Absolute Value, and Number Lines

On a vertical number line representing elevation in meters, a research drone is at coordinate dd and a submarine is at coordinate ss. The coordinate of the drone is a positive integer, and the coordinate of the submarine is a negative integer. The distance between the drone and the submarine is 150150 meters. If the absolute value of the submarine's coordinate is 44 times the coordinate of the drone, what is the coordinate of the drone?

Answer: 30

Answer

The coordinate of the drone is 30.
The correct coordinate of 30 is found by defining the distance between the positive drone coordinate dd and negative submarine coordinate ss as d−s=150d - s = 150. Since s<0s < 0, its absolute value ∣s∣|s| is equal to −s-s. Substituting −s=4d-s = 4d gives d−(−4d)=150d - (-4d) = 150, which simplifies to 5d=1505d = 150 and yields d=30d = 30.

Step-by-Step Solution

1
Set up equations based on the problem description.
d>0d > 0, s<0s < 0, distance ∣d−s∣=150|d - s| = 150, and ∣s∣=4d|s| = 4d.
To translate the verbal description of elevations and distances into mathematical expressions.
2
Simplify the distance and absolute value expressions using the signs of the coordinates.
Since dd is positive and ss is negative, d−s=150d - s = 150. Since ss is negative, ∣s∣=−s|s| = -s, so −s=4d-s = 4d or s=−4ds = -4d.
To eliminate absolute values based on the known signs of the variables.
3
Substitute the expression for ss into the distance equation and solve for dd.
d−(−4d)=150⇒5d=150⇒d=30d - (-4d) = 150 \Rightarrow 5d = 150 \Rightarrow d = 30.
To solve the system of linear equations to find the drone's coordinate.

Key Concept

Using absolute value to represent distance on a number line and solving equations involving sign constraints.
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Topics

Integers, Absolute Value, and Number Lines

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