Problem 17 - Algebra 1: Concepts and Skills

Question

Find the distance between the two points. Round your solution to the nearest hundredth if necessary. $$ (-6,1),(3,1) $$

Step-by-Step Solution

Answer

The distance between the points (-6,1) and (3,1) is 9 units.

Step 1: Identify the given points

The given points are (-6,1) and (3,1). Here, (-6,1) is the first point with x1 = -6 and y1 = 1, and (3,1) is the second point with x2 = 3 and y2 = 1.

Step 2: Apply the distance formula

The formula for the distance between two points in a plane is \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). Since the y-coordinates of the given points are the same, the distance will be the absolute difference in x-coordinates. This can be found by substituting the values of x1 and x2 into the equation: \(d = \sqrt{(3 - (-6))^2 + (1 - 1)^2}\).

Step 3: Calculate the distance

The distance is calculated as \(d = \sqrt{(3 - (-6))^2 + 0}\) which is equal to \(\sqrt{(3 + 6)^2}\) or \(\sqrt{9^2}\). Simplifying gives \(d = 9\).

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that uses a coordinate system to describe and analyze geometric shapes. By plotting points on a coordinate plane, we can explore relationships between different geometric elements.

In a coordinate plane:

• The horizontal axis is called the x-axis.
• The vertical axis is called the y-axis.
• A point is represented by a pair of coordinates (x, y).

These coordinates define the point's exact location in the plane. Coordinate geometry provides tools to calculate distances, angles, and other properties of geometric figures by using algebraic equations. This makes it a powerful tool for solving complex problems easily.

Distance Calculation

Distance calculation is an essential skill in coordinate geometry. To find the distance between two points, we use the distance formula. The formula is derived from the Pythagorean theorem and helps measure the straight-line distance between any two points in a plane.

The distance formula is: \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). This formula computes the distance \(d\) by considering the differences in the x and y-coordinates of the points.

For example, if two points have the same y-coordinates, the distance is simply the absolute difference between the x-coordinates, simplifying the process significantly. In such cases, the distance formula reduces to \(d = |x_2 - x_1|\), making calculations more straightforward.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operators like addition and subtraction. In distance calculations, algebraic expressions help simplify and manipulate the terms involved.

Consider the process involved in finding the distance between the points \((-6, 1)\) and \(3, 1)\). The initial expression is \(\sqrt{(3 - (-6))^2 + (1 - 1)^2}\).

• Calculate \(3 - (-6)\), which simplifies to \(3 + 6\).
• Since the y-coordinates are the same, \(1 - 1 = 0\), making this term zero.

Upon further simplification, this expression results in \(\sqrt{9^2}\), which eventually equals 9. The use of algebra ensures each step is clear and accurate, allowing us to find the correct distance efficiently.