Question

Write the equation of each line in general form. passes through (-3,6)\(;\) is parallel to the \(y\) axis.

Short answer

The equation of the line in general form is x = -3.

Step-by-step solution

Identify the Characteristics of the Line

A line parallel to the y-axis has an undefined slope and does not vary in the x-value. For any given point on this line, the x-coordinate will remain constant.

Determine the Constant Value

Since the line passes through the point (-3,6), the constant x-value for every point on the line will be -3.

Write the Equation in General Form

The general form of the equation of a vertical line is x = a, where a is the constant x-value. Therefore, the equation of our line is x = -3. Since the general form is usually given as Ax + By = C, we can write this as 1x + 0y = -3, which simplifies to x + 0 = -3.

Key concepts

Undefined Slope

When we come across the term undefined slope in coordinate geometry, it refers to the steepness of a line that goes straight up and down, parallel to the y-axis. Imagine climbing a ladder that's placed against a wall; it's a straight-up journey with no movement left or right. In mathematical terms, such a line's slope is considered undefined because we cannot calculate a numerical value for it. Slope is typically represented by the letter m and is calculated by the change in y over the change in x (rise over run). However, for a vertical line, the x-values stay the same while the y-values change, leading to a division by zero situation—which is undefined in mathematics.

Here's why it's important to understand: If you ever encounter an equation that attempts to use slope-intercept form (such as y = mx + b) for a vertical line, you won't be able to find a value for m that makes sense. Instead, these lines are best represented in a different form which we'll discuss in the 'Vertical Line' section.

Vertical Line

A vertical line is a straight line that goes up and down, parallel to the y-axis. No matter how high or low you go on this line, the x-coordinate will always be the same, which is a defining feature of vertical lines. Think of the edge of a ruler or a skyscraper—it doesn't lean left or right. Because of this, vertical lines cannot be represented by the typical y = mx + b equation, since their slope is undefined. Instead, they are described by the equation x = a, where 'a' signifies the constant x-coordinate shared by all points on the line.

In the given exercise, since our line is vertical and passes through the point (-3,6), 'a' is -3. Represented in general form Ax + By = C, we have 1x + 0y = -3, because the line has no change in y, and thus a '0' coefficient for 'y', which essentially removes 'y' from our equation.

Coordinate Geometry

In the exciting world of coordinate geometry, or 'analytic geometry', we study geometric figures using a coordinate system. This usually means working in a two-dimensional plane with perpendicular axes labeled 'x' and 'y'. The magic of coordinate geometry lies in how it connects algebra and geometry, allowing lines, shapes, and curves to be represented and analyzed through equations.

To place this in context, recalling how we map the position of any point with coordinates (x, y), coordinate geometry provides us with tools like the general form of the line's equation (Ax + By = C) to capture the entirety of a line's position and slope within the plane. Whether dealing with a line with a well-defined slope, or the special cases like vertical or horizontal lines with undefined or zero slopes — it all falls under the purview of coordinate geometry, giving us a structured way to comprehend and solve problems involving spatial figures.