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Chapter 1: Problem 39

The coordinates of a particle change by \(\Delta x=5\) and \(\Delta y=6\) as it moves from \(A(x, y)\) to \(B(3,-3) .\) Find \(x\) and \(y .\)

Short Answer

Expert verified
The coordinates are \(x = -2\) and \(y = -9\).

Step by step solution

01

Understand the Change in Coordinates

The particle moves from point \(A(x, y)\) to point \(B(3, -3)\) with a change in coordinates given by \(\Delta x = 5\) and \(\Delta y = 6\). This means that the difference in the x-coordinates is 5 and the difference in the y-coordinates is 6.
02

Set Up Equations for Coordinate Changes

Since the x-coordinate of point B is 3, and the change in x is 5, we set up the equation: \[ x + 5 = 3 \] For the y-coordinate, since the change is from \(y\) to \(-3\) with a change of 6, we set up the equation:\[ y + 6 = -3 \]
03

Solve for the x-coordinate

Solve the equation \( x + 5 = 3 \): Subtract 5 from both sides:\[ x = 3 - 5 \]\[ x = -2 \]
04

Solve for the y-coordinate

Solve the equation \( y + 6 = -3 \): Subtract 6 from both sides:\[ y = -3 - 6 \]\[ y = -9 \]
05

Verify the Solution

Verify the changes by substituting \(x=-2\) and \(y=-9\) back into the change in coordinates:\(\Delta x = (3) - (-2) = 5\), and \(\Delta y = (-3) - (-9) = 6\)Both changes match the given \(\Delta x\) and \(\Delta y\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Coordinate Change
When a particle moves from one point to another in a coordinate plane, the journey it takes is reflected in how its coordinates change. Here, the particle starts at an unknown point \(A(x, y)\) and moves to point \(B(3, -3)\). The provided coordinate changes, \(\Delta x = 5\) and \(\Delta y = 6\), tell us how much the x-value and y-value change during the move. This change can be thought of as:
  • \(\Delta x = x_{\text{final}} - x_{\text{initial}}\)
  • \(\Delta y = y_{\text{final}} - y_{\text{initial}}\)
Given these coordinate changes, we can calculate the unknown starting point by rearranging our coordinate change equations. This is the foundation step for solving more complex problems in coordinate geometry.
Equation Setup
Setting up equations is a crucial step in solving problems involving coordinate geometry. With the changes \(\Delta x = 5\) and \(\Delta y = 6\), we establish relationships between the initial and final coordinates. These equations connect our known and unknown values:
  • For the x-coordinate: \(x + 5 = 3\)
  • For the y-coordinate: \(y + 6 = -3\)
By understanding these changes, we can set up equations that capture the transition from point A to B. The equations reflect a simple algebra problem that, once solved, reveals the initial x and y values.
Coordinate Solution
Once the equations are set up, solving them becomes a straightforward process. We focus on finding the values of \(x\) and \(y\):
  • Solve for x: Using the equation \(x + 5 = 3\), we isolate \(x\) by subtracting 5 from both sides, yielding \(x = -2\).
  • Solve for y: Similarly, from the equation \(y + 6 = -3\), subtracting 6 from both sides gives us \(y = -9\).
This approach illustrates how solving these types of equations helps find precise initial coordinates. By tackling one equation at a time, we ensure clarity and accuracy in reaching the correct solution.
Particle Motion
The concept of particle motion in coordinate geometry is driven by understanding how a particle's coordinates transform as it moves. As the particle traveled from point A to B with given changes \(\Delta x = 5\) and \(\Delta y = 6\), we consider this motion as a journey of shifts in position:
  • The x-coordinate changes by moving horizontally 5 units from the start.
  • The y-coordinate shifts by moving vertically 6 units.
By tracking these changes, the motion of the particle is viewed through its change in position. Understanding this concept is essential in predicting and estimating further movement in the problem-solving process.

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