Problem 8 - College Algebra Essentials

Question

find the distance between each pair of points. If necessary, round answers to two decimals places. $$ (-4,-1) \text { and }(2,-3) $$

Step-by-Step Solution

Answer

The distance between the points (-4,-1) and (2,-3) is approximately equal to \( \sqrt{40} \), which is roughly 6.32 (rounded to two decimal places).

Step 1: Identify the coordinates

First, identify the Cartesian coordinates (x, y) of the two points. The first point has coordinates (-4, -1) and the second point has coordinates (2, -3). So, \(x_1 = -4\), \(y_1 = -1\), \(x_2 = 2\), and \(y_2 = -3\).

Step 2: Substitute values into the formula

Next, substitute these values into the formula for the distance d between two points. That gives \[ d = \sqrt{(2 - (-4))^2 + (-3 - (-1))^2} \]

Step 3: Simplify

Simplify the equation to get an expression that can be calculated: \[ d = \sqrt{(6)^2 + (-2)^2} = \sqrt{36 + 4} \]

Step 4: Calculate the distance

Calculate the final distance d: \[ d = \sqrt{40} \]

Key Concepts

Understanding Cartesian Coordinates

The concept of Cartesian coordinates is essential in mathematics, particularly in geometry and algebra. These coordinates allow us to pinpoint the exact location of a point on a two-dimensional plane. In the Cartesian coordinate system, each point is defined by a pair of numbers written as \(x, y\).

• The first number, \(x\), is the horizontal position on the x-axis.
• The second number, \(y\), is the vertical position on the y-axis.

Knowing the coordinates \((-4, -1)\) and \( (2, -3) \) allows us to represent these points on a graph. This system is fundamental in calculating the distance between such points, which we can achieve using the distance formula.

Squaring Differences

The next step in finding the distance between two points involves squaring the differences of their coordinates. This is derived from the Pythagorean theorem. When you subtract corresponding coordinates \(x_2 - x_1\) and \(y_2 - y_1\), you obtain the difference in the horizontal and vertical distances between the points. Squaring these differences has several purposes:

• It eliminates any negative signs that may result from subtraction.
• It is a step toward calculating the actual distance, ensuring all values become positive contributions.

In our example, the calculations were: \( (2 - (-4))^2 \) which simplifies to \(6^2\) and \((-3 - (-1))^2 \) simplifying to \((-2)^2\)."

Square Root Calculation

After obtaining the squared differences, the following step involves calculating the square root. This operation is essential because it converts the squared units back into linear distance, making it meaningful.

• When applying the square root, the goal is to derive a single numeric distance.
• It reverses the earlier squaring process and provides a more intuitive measure of distance.

In the problem, \( \sqrt{36 + 4}\ \) returns the true linear distance between the points. This step brings our calculations closer to finding the answer.

Simplifying Expressions

The last phase of this distance calculation involves expression simplification. This streamlines the process by reducing the equation to its simplest form, preparing it for final calculation.

• Combining like terms ensures the expression contains only necessary components.
• It makes the final operations understandable and manageable.

Simplification in our context led to \( \sqrt{40} \), simplifying the calculation by reducing intermediate steps. The last part might involve rounding or further calculations but keeping it simplified at this stage sets up for easy computation.