Problem 40 - Algebra 1

Question

Without calculating, state whether the slope of the line through the points is positive, negative, zero, or undefined. (2,5),(4,1)

Step-by-Step Solution

Answer

The slope of the line through the points (2,5) and (4,1) is negative.

Step 1: Identify the Given Points

The two points given are (2,5) and (4,1). The first value in each pair represents the x-coordinate, and the second value represents the y-coordinate.

Step 2: Compare the 'y' values

In moving from point (2,5) to (4,1), observe the change in Y-coordinates. The Y-coordinate decreases, going from 5 in the first point down to 1 in the second point.

Step 3: Determine the Slope Type

Since the y-value decreases from the first point to the second, the slope of the line through these points is negative.

Key Concepts

Coordinates

Coordinates are used to describe the position of a point on a plane. In the Cartesian plane, each point is expressed as a pair of numbers: the x-coordinate and the y-coordinate.
The x-coordinate, or abscissa, indicates the horizontal position, while the y-coordinate, or ordinate, indicates the vertical position.

• For example, in the point (2,5), 2 is the x-coordinate, and 5 is the y-coordinate.
• A positive x-coordinate means the point is to the right of the y-axis, while a positive y-coordinate means the point is above the x-axis.
• Conversely, negative values indicate positions left of the y-axis or below the x-axis.

Understanding coordinates is crucial for graphing points, plotting lines, and analyzing geometric relationships.

Positive Slope

A positive slope indicates that as you move from left to right along the line, the line rises. This means the y-coordinates of the points increase as the x-coordinates increase.

• In mathematical terms, if you have two points, \((x_1, y_1)\) and \((x_2, y_2)\), the slope is calculated as \( \frac{y_2 - y_1}{x_2 - x_1} \).
• If this fraction results in a positive number, the slope of the line is positive.

For example, consider the points (3, 2) and (5, 6). As you move from the first point to the second, the y-value increases, indicating a positive slope. This means the line through these points goes upwards.

Negative Slope

A negative slope means that the line descends as you move from left to right. This is seen when the y-coordinates of points on the line decrease as the x-coordinates increase.
For any two points, calculate the slope using: \( \frac{y_2 - y_1}{x_2 - x_1} \).

• A negative result indicates a negative slope.

In the exercise example, the points are (2,5) and (4,1).

• The y-coordinate decreases from 5 to 1, while the x-coordinate increases from 2 to 4, resulting in a negative slope.

This means the line through these points slopes downwards.

Undefined Slope

An undefined slope occurs when a line is vertical. This happens when two points have the same x-coordinate, but different y-coordinates.
For example, consider points like (3,2) and (3,6).

• The x-coordinates are the same and the y-coordinates are different, indicating a vertical line.
• The slope formula \( \frac{y_2 - y_1}{x_2 - x_1} \) would involve division by zero, which is undefined in mathematics.

Therefore, any vertical line has an undefined slope because it doesn't tilt to the right or left, it goes straight up and down.