Problem 18 - Thinking Mathematically

Question

Calculate the slope of the line passing through the given points. If the slope is undefined, so state. Then indicate whether the line rises, falls, is horizontal, or is vertical. \((3,0)\) and \((0,-9)\)

Step-by-Step Solution

Answer

The slope of the line passing through the points \((3,0)\) and \((0,-9)\) is \(3\). The line rises.

Step 1: Determine the Coordinates

First, assign names to the given coordinates: let's call \((3,0)\) as 'Point 1' and \((0,-9)\) as 'Point 2'.

Step 2: Calculate the Slope

The formula to calculate the slope \(m\) when two points \((x_1, y_1)\) and \((x_2, y_2)\) are given is given by the equation \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Substituting the given points of our problem into this formula we get \(m = \frac{-9 - 0}{0 - 3} = 3\).

Step 3: Interpreting the Slope

A positive slope means the line rises as it moves from left to right. In this case, the slope is \(3\) which is positive. Hence, it can be said that the line rises.

Key Concepts

Slope of a Line

When learning about linear relationships, understanding the slope is crucial. It measures the steepness of a line, defined as the ratio of the vertical change (rise) to the horizontal change (run) between two distinct points on the line. The slope is often represented by the letter 'm'. The basic formula to calculate slope is:
\[ m = \frac{\text{change in y}}{\text{change in x}} = \frac{y_2 - y_1}{x_2 - x_1} \]
In practice, to find the slope like in our exercise, locate two points on the line. We'll designate these as \((x_1, y_1)\) and \((x_2, y_2)\). Using the slope formula, we subtract the y-coordinate of the first point from the y-coordinate of the second point, and place this difference above the difference of the x-coordinates, calculated In the same way, in a fraction.

Understanding Slope Values

A positive slope indicates the line rises from left to right, while a negative slope shows it falls. A slope of zero corresponds to a horizontal line, and an undefined slope (where we try to divide by zero because the run is zero) means the line is vertical. Therefore, if we observe an expression such as \(m = 3\), it reveals a line that consistently rises, increasing three units vertically for every single unit it moves horizontally.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, blends algebra and geometry to describe the positions of points, lines, and figures on a coordinate plane. A coordinate system consists typically of two axes at right angles to each other, forming a plane divided into four quadrants. Each point on this plane is represented by an ordered pair of numbers, often denoted as \((x, y)\), where 'x' stands for the horizontal position, and 'y' for the vertical.

Identifying Points

To identify the position of any point, we start at the origin (where both x and y are zero), then move along the x-axis by the amount specified by the first number, and proceed parallel to the y-axis by the second number. The exercise given demonstrates this, where we have \((3, 0)\) and \((0, -9)\). The first point lies on the positive side of the x-axis while the second point falls on the negative side of the y-axis.

• Horizontal lines have equal y-coordinates and an undefined x-coordinate variation.
• Vertical lines have equal x-coordinates and an undefined y-coordinate variation.

Linear Equations

Linear equations form the foundation of linear relationships in algebra. They represent straight lines when plotted on a coordinate plane and can be presented in various forms, such as the slope-intercept form: \(y = mx + b\), where 'm' is the slope, and 'b' is the y-intercept, the point where the line crosses the y-axis.

Interpreting Linear Equations

Every solution of a linear equation is a point that lies on the line represented by the equation. In contrast, a pair of linear equations might intersect (one solution), be parallel (no solution), or coincide (infinitely many solutions). The important qualities of a linear equation are its slope and y-intercept as they dictate the angle of inclination and the starting point of the line respectively. For instance, an equation like \(y = 3x - 2\) has a slope of 3 and a y-intercept of -2, which means the line rises three units vertically for every horizontal unit and begins two units below the origin on the y-axis.