Problem 6 - College Algebra

Question

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

Step-by-Step Solution

Answer

The distance between the points (0,0) and (3,-4) is 5 units.

Step 1: Identify the given points

The two points given are \((0,0)\) and \((3,-4)\). The coordinates for the first point, \((0,0)\), are \(x_1 = 0\) and \(y_1 = 0\). The coordinates for the second point, \((3,-4)\), are \(x_2 = 3\) and \(y_2 = -4\).

Step 2: Substitute into the Distance formula

Next substitute the identified coordinates into the distance formula \(d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\). This gives \(d = \sqrt{(3-0)^2 + (-4-0)^2} = \sqrt{(3)^2 + (-4)^2}\).

Step 3: Simplify and Compute

Now square the differences, and then add the squares: \(d = \sqrt{3^2 + -4^2} = \sqrt{9 + 16} = \sqrt{25}\). The last step is to take the square root of 25, which gives \(d = 5\).

Key Concepts

Cartesian Coordinates

In mathematics, the Cartesian coordinate system is a two-dimensional plane where each point is determined by an ordered pair of numbers. These numbers are known as the coordinates, written as \((x, y)\). The first number, \(x\), represents the horizontal position, while the second number, \(y\), indicates the vertical position. The origin of the system is the point \((0, 0)\), where the horizontal and vertical axes intersect.
Understanding this system is essential for locating points on the plane. For instance, the first point \((0, 0)\) stays at the origin, and the second point \((3, -4)\) is three units to the right and four units down.

• The Cartesian plane consists of two axes: the x-axis (horizontal) and the y-axis (vertical).
• Coordinates are written as \((x, y)\), with \(x\) determining the position along the x-axis and \(y\) determining the position along the y-axis.
• Points can have positive or negative values based on their position relative to the origin.

Understanding how to plot and interpret these coordinates lays the groundwork for more complex mathematical concepts, such as distance calculation between two points.

Distance Calculation

The distance formula is a handy tool for finding the distance between two points on the Cartesian plane. It stems from the Pythagorean theorem, which relates the sides of a right triangle: \(a^2 + b^2 = c^2\).
To find the distance between two points \((x_1, y_1)\) and \((x_2, y_2)\), we use the formula \(d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\).

• First, calculate the difference between the x-coordinates: \(x_2 - x_1\).
• Next, calculate the difference between the y-coordinates: \(y_2 - y_1\).
• Square these differences to ensure they are non-negative.
• Add the squared values together and take the square root of the sum.

Applying this formula helps us measure straight-line distance between any two points, which is essential in various fields like physics, geography, and even art.

Mathematical Computation

To effectively use mathematical formulas like the distance formula, it's important to break down each computation into manageable steps. Let's walk through the procedure again, using our points \((0, 0)\) and \((3, -4)\):

• Identify the points, knowing \(x_1 = 0\), \(y_1 = 0\) and \(x_2 = 3\), \(y_2 = -4\).
• Plug these values into the formula: \(d = \sqrt{(3-0)^2 + (-4-0)^2}\).
• Calculate: \((3-0)^2 = 9\) and \((-4-0)^2 = 16\).
• Add these results: \(\sqrt{9 + 16} = \sqrt{25}\).
• Finally, compute the square root: \(d = 5\).

This step-by-step approach helps in understanding how each part of the formula contributes to the final result. Practicing mathematical computations not only enhances problem-solving skills but also strengthens the ability to work through complex problems methodically.