Problem 1 - Understanding Elementary Algebra with Geometry

Question

Find the slope of the line passing through the given points. Round to the nearest hundredth where necessary. \((3,5)\) and \((6,9)\)

Step-by-Step Solution

Answer

The slope of the line is approximately 1.33.

Step 1: Identify the Given Points

The given points are \(3, 5\) and \(6, 9\). Label the coordinates as follows: \(x_1 = 3, y_1 = 5\) and \(x_2 = 6, y_2 = 9\).

Step 2: Apply the Slope Formula

To find the slope \(m\) of the line passing through the points, use the formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Substitute the values: \( m = \frac{9 - 5}{6 - 3} \).

Step 3: Simplify the Expression

Simplify the numerator and the denominator: \( m = \frac{4}{3} \).

Step 4: Round to the Nearest Hundredth

Converting \( \frac{4}{3} \) to a decimal gives approximately 1.3333. Rounding to the nearest hundredth, the slope is approximately 1.33.

Key Concepts

Coordinate Geometry

Coordinate Geometry is a branch of mathematics that uses the coordinate system to represent and solve geometric problems. In this system, each point is defined by an ordered pair of numbers, which are its coordinates. The coordinates \(x\) and \(y\) specify the position of a point in the plane. For example, the point \(3, 5\) means that it is located 3 units along the x-axis and 5 units up the y-axis. This system allows us to easily visualize and compute various properties of geometric shapes.

Slope Formula

The Slope Formula is a mathematical way to find the steepness or inclination of a line. It comes in handy when you want to know how much a line rises or falls as you move from one point to another. The slope \(m\) of a line passing through points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by the formula \[ m = \frac{y_2 - y_1}{x_2 - x_1} \].

• \ y_2 \ and \ y_1 \ are the y-coordinates of the two points
• \ x_2 \ and \ x_1 \ are the x-coordinates.

Make sure to subtract the y-coordinates to get the numerator and the x-coordinates to get the denominator. The result is the slope, which can tell us whether the line is rising, falling, or horizontal.

Linear Equations

Linear Equations are equations that describe a straight line in coordinate geometry. They are typically written in the form \[ y = mx + c \], where:

• \ y \ represents the y-coordinate
• \ x \ represents the x-coordinate
• \ m \ is the slope of the line
• \ c \ is the y-intercept, the value of y when x = 0

In the given exercise, we are asked to find the slope of the line through specific points. Once you have the slope using the slope formula, you can even plug it into this linear equation to describe the entire line. Understanding linear equations is crucial for solving higher-level problems in coordinate geometry.