Question

In Exercises \(21-24,\) write a general linear equation for the line through the two points. $$(-2,0), \quad(-2,-2)$$

Short answer

The equation of the line that passes through the points (-2,0) and (-2,-2) is \(x = -2\).

Step-by-step solution

Determine the form of the equation

Recognize that this is a vertical line, and the equation for vertical lines is always \(x = k\), where \(k\) is the x-coordinate of the points on the line. We don't need to find the slope or y-intercept.

Substitute the x-coordinate

From either of our points, we can see that the x-coordinate is -2. Thus, the equation of the line becomes \(x = -2\).

Write the final equation

The final equation is \(x = -2\). This equation implies that every point on this line has an x-coordinate of -2, no matter what the y-coordinate is, which we can see is true from the original points we were given.

Key concepts

Equation of a Vertical Line

When we talk about the equation of a vertical line in the context of a coordinate plane, we're referring to a line that goes straight up and down. Unlike horizontal lines, vertical lines cannot be expressed using the typical slope-intercept form of a linear equation, which is \(y = mx + b\). That's because the slope (\(m\)) of a vertical line is undefined. Instead, all points on a vertical line have the same \(x\)-coordinate.

The standard equation for a vertical line is simply \(x = k\), where \(k\) is the constant \(x\)-coordinate for all points on the line. This makes graphing a vertical line quite straightforward; you just draw a line through the specified \(x\)-coordinate on the graph, and it will extend infinitely in both the positive and negative \(y\)-directions.

For example, if you have a vertical line where every point has an \(x\)-coordinate of -2, the equation is \(x = -2\). No matter what values the \(y\)-coordinates take, the line will not deviate from this vertical path.

X-Coordinate

The \(x\)-coordinate is a fundamental concept in the realm of coordinate geometry. It tells us the horizontal position of a point on the coordinate plane, with respect to the origin. The \(x\)-axis itself is a horizontal line that helps us determine the position of the point. Positive \(x\)-coordinates indicate positions to the right of the origin, while negative \(x\)-coordinates are found to the left.

When dealing with points on a vertical line, all the points share the same \(x\)-coordinate because they all fall directly above and below each other. This shared \(x\)-coordinate becomes the defining feature of the line's equation. In our textbook exercise example, the \(x\)-coordinate for both points \( (-2, 0) \) and \( (-2, -2) \) is -2, which is why the equation of the line that passes through these points is \(x = -2\).

Graphing Lines in the Coordinate Plane

Graphing lines on the coordinate plane is a core skill in algebra and geometry. The coordinate plane consists of two perpendicular number lines: the horizontal \(x\)-axis and the vertical \(y\)-axis. The intersection of these axes forms the origin, which has coordinates \( (0, 0) \).

To graph a line, you need either two points, a point and the slope, or an equation. Horizontal and vertical lines are the exceptions, as they do not have a definable slope in the traditional sense. For vertical lines, since the \(x\)-coordinate is constant, you can plot two points with that \(x\)-value and draw a line through them. It will intersect the \(x\)-axis at that \(x\)-value and will never touch the \(y\)-axis, reflecting the fact that \(x\) doesn’t change, but \(y\) can be any value.

For the given exercise, graphing \(x = -2\) means plotting points along the line where \(x\) is always -2. This creates a perfectly vertical line that illustrates the unique relationship between points on such a line and their constant \(x\)-coordinate.