Problem 3 - College Algebra

Question

Find the distance between each pair of points. If necessary, round answers to two decimals places. $$(4,-1)\( and \)(-6,3)$$

Step-by-Step Solution

Answer

The distance between the points (4,-1) and (-6,3) is approximately 11.66 units.

Step 1: Understanding the Given Coordinates

Identify the coordinates of the two points. Here the points are given as (4,-1) and (-6,3). The coordinates of the first point are (x1, y1) = (4, -1) and the second point are (x2, y2) = (-6, 3).

Step 2: Apply the Distance Formula

The distance, d, between two points in a Cartesian coordinate system is given by the formula: \[d = \sqrt{{(x2 - x1)^2 + (y2 - y1)^2}}\]

Step 3: Plug in the values and calculate the distance

Substitute the actual values into the distance formula and perform the calculation: \[d = \sqrt{{(-6 - 4)^2 + (3 - (-1))^2}} \approx 11.66\] It is necessary to obey the order of operations, meaning that operations in brackets are calculated first, then the exponentiation (squaring), then addition, and finally the square root.

Key Concepts

Cartesian coordinates

Cartesian coordinates are a fundamental concept in mathematics used to describe the location of a point in a plane. Each point is defined by a pair of numerical coordinates. These coordinates tell you how far along a line from the origin (0, 0) a point is located. In this system:

• The horizontal distance is described by the x-coordinate.
• The vertical distance is described by the y-coordinate.

In the given exercise, we had two points:

• Point A with coordinates (4, -1)
• Point B with coordinates (-6, 3)

Here, (4, -1) means move 4 units right and 1 unit down from the origin, while (-6, 3) means move 6 units left and 3 units up. Understanding how to locate points using Cartesian coordinates is crucial for performing further operations such as distance calculations.
Carrying out calculations using these coordinates is an essential step in the context of geometry and graph-related problems.

distance calculation

Distance calculation in geometry refers to finding the length between two points in space. When we talk about points in a Cartesian plane, we use the distance formula, derived from the Pythagorean theorem:
\[d = \sqrt{{(x_2 - x_1)^2 + (y_2 - y_1)^2}}\]
This formula helps compute the distance between two points by taking the differences between the x and y coordinates, squaring them, adding them together, and finally taking the square root to get the Euclidean distance. This is because, on a plane, the distance represents forming a hypotenuse in a right triangle whose sides are parallel to the axes.
In the exercise, using the formula for points (4, -1) and (-6, 3):

• Subtract the x-coordinates: (-6 - 4) = -10
• Subtract the y-coordinates: (3 - (-1)) = 4
• Square the results and add them: d^2 = (-10)^2 + 4^2 = 100 + 16 = 116
• Take the square root: d = \sqrt{116} \approx 11.66

The distance between these points is approximately 11.66 units.

order of operations

Following the correct order of operations is crucial in mathematics to get the right answer. When performing arithmetic calculations like those used in the distance formula, follow this order:

• First, calculate any operations inside parentheses.
• Then, perform exponentiation (like squaring a number).
• Next, handle any multiplication or division.
• Finally, complete any addition or subtraction.

For our problem, here's how this applies: 1. Inside the distance formula, the expressions (x_2 - x_1) and (y_2 - y_1) should be simplified first. 2. Next, square these simplified results. 3. Add the squared values. 4. Finally, use the square root to find the distance. For the given coordinates, calculate: - Calculate (-6 - 4) and (3 - (-1)). - Square each result. - Add the squared numbers together. - Then take the square root of this sum. Respecting this order ensures that each part of the formula is evaluated correctly, ultimately giving you the correct distance.