Problem 9 - College Algebra Essentials

Question

find the distance between each pair of points. If necessary, round answers to two decimals places. $$ (0,-3) \text { and }(4,1) $$

Step-by-Step Solution

Answer

The distance between the points (0,-3) and (4,1) is approximately 5.66.

Step 1: Identify the Coordinates

The coordinates of the two points are (0,-3) and (4,1). So, \(x1 = 0\), \(y1 = -3\), \(x2 = 4\), and \(y2 = 1\).

Step 2: Substitute the Coordinates into the Formula

Substitute the coordinates into the formula for calculating distance. The formula is \(d = \sqrt{(x2-x1)^2 + (y2-y1)^2}\). Plug in the values and we get \(d = \sqrt{(4-0)^2 + (1-(-3))^2}\).

Step 3: Simplify the Equation

Simplify the equation to \(d = \sqrt{(4)^2 + (4)^2}\). Following through with the simplification further, we get \(d = \sqrt{(16) + (16)}\), and finally \(d = \sqrt{32}\).

Step 4: Evaluate the Result

Evaluate the square root of 32 to find the final distance. If needed, round the answer to two decimal places. With a calculator, we find that \(\sqrt{32}\) is approximately equal to 5.66.

Key Concepts

Coordinate Geometry

Coordinate geometry is the study of geometric figures through a coordinate system. Picture a graph with an x-axis (horizontal) and a y-axis (vertical) that intersect at zero, creating a plane known as the Cartesian plane. Each point on this plane is specified by a pair of numbers called coordinates. The coordinates are written as \(x, y\), where \(x\) represents the horizontal position and \(y\) the vertical position. For example, the point (0, -3) is located at zero in the x-direction and negative three in the y-direction.
Understanding coordinate geometry helps us visualize points and distances. The plane provides a convenient way to perform calculations and solve geometric problems algebraically, such as finding the distance between two given points.

Distance Calculation

The distance formula comes in handy when we need to calculate how far apart two points are on a coordinate plane. Derived from the Pythagorean theorem, this formula is presented as \(d = \sqrt{(x2-x1)^2 + (y2-y1)^2}\). It calculates the straight-line distance between any two points \((x1, y1)\) and \((x2, y2)\).

For instance, given the points \( (0, -3) \) and \((4, 1)\), we substitute these coordinates into the formula. Here's how you do it:

• Subtract the x-coordinates: \(4 - 0 = 4\)
• Subtract the y-coordinates: \(1 - (-3) = 4\)
• Square these results: \(4^2 = 16\) and \(4^2 = 16\)

Add these squared numbers to get \(16 + 16 = 32\). Finally, take the square root of 32 to get approximately 5.66, which is the distance between the two points.

Algebraic Problems

Algebraic problems often require solving for unknowns or simplifying equations. In terms of geometry, it's about applying algebraic methods to find lengths, distances, or angles on the coordinate plane. Using formulas, like the distance formula, requires substituting known values and performing arithmetic operations.
To tackle these kinds of problems efficiently:

• Identify the coordinates and assign them variables, like \((x1, y1)\) and \((x2, y2)\)
• Substitute these into an established formula
• Simplify step by step, using arithmetic and algebraic properties

This structured approach not only helps solve the problem at hand but also deepens your understanding of how algebra and geometry interconnect. Whether solving for a length or distance, precision in these calculations is crucial for getting a correct and accurate result.