Problem 45 - Understanding Elementary Algebra with Geometry

Question

Sketch the graph of the line satisfying the given conditions. Passing through \((-1,-5)\) and whose slope is undefined

Step-by-Step Solution

Answer

The line is vertical at x = -1.

Step 1: Identify the Slope

An undefined slope indicates a vertical line. Vertical lines have the same x-coordinate for all points.

Step 2: Determine the X-Coordinate

Given the line passes through (-1, -5), the x-coordinate for the vertical line is -1.

Step 3: Write the Equation

Since it's a vertical line with x-coordinate -1, the equation of the line is \(x = -1\).

Step 4: Sketch the Graph

Draw a vertical line on the coordinate plane at x = -1. This line extends indefinitely in the y-direction.

Key Concepts

undefined slope

When we talk about an undefined slope, we are referring to the slope of a vertical line. In a vertical line, the change in the y-coordinates can be any value, but the change in the x-coordinates is zero.

The slope formula is \(\frac{\text{rise}}{\text{run}}\). For a vertical line, the 'run' part (change in x) is 0, leading to \(\frac{\text{rise}}{0}\), which is undefined.

This is because we cannot divide by zero in mathematics. Therefore, a vertical line has an undefined slope. If you see an undefined slope, always think of a vertical line.

vertical line equation

The equation of a vertical line is very simple. It is always in the form of \(x = a\), where 'a' is the x-coordinate for all the points on the line.

In the given exercise, the line passes through the point \((-1, -5)\). This means every point on this line has an x-coordinate of -1. Thus, the equation of this vertical line is \(\boxed{x = -1}\).

This equation tells us that no matter what the y-coordinate is, the x-coordinate will always be -1. This is a defining feature of vertical lines.

coordinate plane

In order to sketch graphs, especially vertical lines, understanding the coordinate plane is crucial.

The coordinate plane has two axes: the x-axis (horizontal) and the y-axis (vertical). They intersect at a point called the origin, \((0,0)\).

Each point on the plane is represented by an ordered pair \((x,y)\), where 'x' is the horizontal distance from the origin and 'y' is the vertical distance.

To graph a vertical line such as \(\boxed{x = -1}\), you:

• Identify the value on the x-axis (-1 in this case).
• Draw a straight line passing through all points where x is -1.

This vertical line will extend infinitely in both the positive and negative y-directions.