Problem 14 - 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. \((6,-4)\) and \((4,-2)\)

Step-by-Step Solution

Answer

The slope of the line passing through the points (6,-4) and (4,-2) is -1. Thus, the line falls from left to right.

Step 1: Identify the coordinates of the points

The two points provided are (6,-4) and (4,-2). We can treat (6,-4) as (x1, y1) and (4,-2) as (x2, y2).

Step 2: Substitute the coordinates into the slope formula

By substituting the coordinates into the formula, \(m = \frac{-2 - (-4)}{4 - 6} = \frac{2}{-2}\).

Step 3: Compute the slope

After calculating the fraction, we find that the slope \(m = -1\).

Step 4: Interpret the slope

Because the slope is negative, we can say that the line falls from left to right.

Key Concepts

Slope of a line

The slope of a line is a measure that tells us how steep a line is. In algebra, it reflects how much the line goes up or down as you move from one point to another. The slope is typically represented by the letter \( m \) and can be calculated using the formula:
\[m = \frac{y_2 - y_1}{x_2 - x_1}\]This formula requires two points on the line, denoted as \((x_1, y_1)\) and \((x_2, y_2)\). It tells us the change in the y-coordinates divided by the change in the x-coordinates. If the slope is:

• Positive, the line rises.
• Negative, the line falls.
• Zero, the line is horizontal.
• Undefined, the line is vertical.

Understanding slope is crucial for graphing lines and analyzing their behavior.

Coordinates

Coordinates are a pair of numbers that specify the position of a point in a two-dimensional space. They are written in the form \((x, y)\), where \(x\) is the horizontal position, and \(y\) is the vertical position.
In our problem, we identified two points: \((6, -4)\) and \((4, -2)\). These points represent locations on a plane. They allow us to calculate the slope and understand geometric shapes and patterns.
Keep in mind:

• The first number is always the x-coordinate.
• The second number is the y-coordinate.

Coordinates play a fundamental role in geometry and algebra, helping define shapes and solving equations.

Interpreting slope

Interpreting the slope involves understanding its direction and meaning in context. A slope tells us not only the steepness but also the direction of a line. Here's how to interpret it:

• A slope of \(-1\) indicates that for every unit we move right, the line moves down one unit. This is why we describe the line as falling.
• If the slope were positive, the line would be rising. For instance, a slope of \(2\) means that for every unit right, the line goes up by two units.
• A horizontal line has a slope of \(0\), while a vertical line has an undefined slope.

The sign and value of the slope are integral in discerning the nature of a line's path across a coordinate plane.

Point-slope formula

The point-slope formula is a powerful tool in algebra to write the equation of a line when the slope \(m\) and one point \((x_1, y_1)\) are known. The formula is:
\[y - y_1 = m(x - x_1)\]This allows us to express the relationship between x and y on the line. Say we have a line with a slope of \(-1\) passing through \((6, -4)\). We can plug these values into the formula:
\[y - (-4) = -1(x - 6)\]Simplifying gives us the equation of the line:
\[y + 4 = -x + 6\]
which further simplifies to \[y = -x + 2\].
This way, we can understand how the line behaves and predict the y-coordinate for any given x-coordinate.