Problem 18 - Algebra 1

Question

Find the distance between the two points. Round the result to the nearest hundredth if necessary. 18\. $$(-3,1),(2,6)$$

Step-by-Step Solution

Answer

The distance between the points (-3,1) and (2,6) rounded to the nearest hundredth is 7.07.

Step 1: Identify the coordinates

From the given problem, we can see that the coordinates are (-3,1) and (2,6), identifying them to be (x_1, y_1) = (-3,1) and (x_2, y_2) = (2,6) respectively.

Step 2: Substitute into the distance formula

Substitute the known values into the formula \(d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\), resulting in \(d = \sqrt{(2-(-3))^2 + (6-1)^2}\).

Step 3: Simplify the expression

Simplify the expression to get \(d = \sqrt{(5)^2 + (5)^2}\) which results in \(d = \sqrt{50}\). This would round to about 7.07 when rounded to the nearest hundredth.

Key Concepts

Understanding Coordinates

When addressing the concept of coordinates, envision a grid system, like a map, which allows us to pinpoint specific locations. In a two-dimensional plane, a coordinate is written as a pair \( (x, y) \), where \( x \) represents the position along the horizontal axis and \( y \) signifies the location on the vertical axis. The first value \( x \) is called the abscissa, and the second value \( y \) is known as the ordinate.

Applying this to the exercise, the coordinates given are \( (-3,1) \) and \( (2,6) \) which indicates two distinct points on the plane. To find the distance between these points, these coordinates help us comprehend how far apart they lie horizontally and vertically. Within the context of this problem, think of these coordinates as the starting and ending points of a journey on this grid, where the distance formula will be the guide to measure the exact path between them.

How to Substitute Values

Substituting values is akin to replacing placeholders with actual data. The distance formula, \( d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} \), contains placeholders \( x_1 \)\( x_2 \)\( y_1 \) and \( y_2 \) that correspond to the coordinates of our points. When we substitute values, we carefully replace these placeholders with the appropriate numbers from our given coordinates.

In the exercise, we have the coordinates \( (-3,1) \) and \( (2,6) \) which are our \( x_1 \)\( y_1 \) and \( x_2 \)\( y_2 \) respectively. The actual substitution looks like this: \( d = \sqrt{(2-(-3))^2 + (6-1)^2} \). This process is crucial because correct substitution prevents errors in the subsequent steps and ensures that the computation accurately reflects the distance between the two points.

Simplifying Mathematical Expressions

Simplifying expressions means to perform all possible arithmetic operations to reduce the expression to its simplest form. It's important because it makes the expression easier to read and understand. Once you have substituted values into the distance formula, the next step is to rationalize and condense the expression further.

For our exercise's expression \( d = \sqrt{(2-(-3))^2 + (6-1)^2} \), the initial simplification involves carrying out the operations within the parentheses: \( d = \sqrt{(5)^2 + (5)^2} \). Subsequent simplification includes squaring the numbers inside the radical to get \( d = \sqrt{50} \) and then finding the square root of that sum. For students, understanding how to simplify expressions is essential for solving problems efficiently and correctly as it is a foundational skill for all areas of mathematics.