Problem 32 - Algebra 1

Question

Plot the points and find the slope of the line passing through the points. $$\left(0, \frac{1}{2}\right),(0,0)$$

Step-by-Step Solution

Answer

The slope of the line passing through the points (0, 1/2) and (0, 0) is undefined, indicating a vertical line.

Step 1: Plotting Points

Begin by plotting the points (0, 1/2) and (0, 0) on the coordinate plane. As both x-coordinates are 0, these two points lie on the y-axis.

Step 2: Calculating Slope

To calculate the slope (m) of the line passing through these two points, use the formula m = (y2 - y1) / (x2 - x1). Substituting the given points into this formula gives: m = (0 - 1/2)/(0 - 0). This results in a division by zero situation, which is undefined.

Step 3: Interpreting the Result

When the slope of a line is undefined, it means the line is vertical. In the cartesian plane, a vertical line goes straight up and down and has no slope. Therefore the line passing through points (0, 1/2) and (0,0) is a vertical line.

Key Concepts

Coordinate plane

The coordinate plane is a foundational concept in understanding geometry and algebra.It is a two-dimensional plane formed by the intersection of a horizontal axis, labeled the x-axis, and a vertical axis, called the y-axis.Where these two axes intersect is known as the origin, represented as the point (0, 0).
The coordinate plane allows us to locate points using pairs of numbers, known as coordinates.Coordinates are written as ordered pairs \((x, y)\), where the first number represents the horizontal position (x-coordinate), and the second number indicates the vertical position (y-coordinate).Using this system, any point on the plane can be precisely defined.
To plot a point, you start at the origin. Move along the x-axis to your desired x-coordinate, and from there, move parallel to the y-axis to place your point at the correct y-coordinate.This method of plotting points is crucial for visualizing and solving many mathematical problems.

Undefined slope

When talking about the slope of a line, we refer to the steepness or the incline of the line. The slope is typically calculated using the formula \(m = \frac{{y_2 - y_1}}{{x_2 - x_1}}\).The variables \(y_2, y_1\) are the y-coordinates of two points on the line, and \(x_2, x_1\) are their corresponding x-coordinates.
An undefined slope occurs when the denominator of this formula, \(x_2 - x_1\), equals zero.This situation arises when both points on the line share the same x-coordinate.As division by zero is undefined in mathematics, the slope of such a line cannot be calculated using this formula.Thus, the slope is considered to be undefined.
An undefined slope indicates a vertical line on the coordinate plane, which leads us to our next concept.

Vertical line

A vertical line is a line that runs straight up and down on the coordinate plane.Such a line can be drawn through any point with the same x-coordinate.For instance, any line passing through points like (0, 1/2) and (0, 0) will be vertical.
Vertical lines have some unique characteristics:

• They do not have a slope. Instead, their slope is undefined, as discussed previously.
• They intersect the x-axis exactly once and can continue infinitely above and below that intersection point.
• The equation of a vertical line is always in the form of \(x = a\), where "a" is the constant x-coordinate of the points on the line.

Understanding vertical lines is essential as they often appear in practical problems and in the study of functions and relationships in mathematics.