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Chapter 3: Problem 105

Write an equation of the line parallel to the \(y\) -axis and passing through \((-2,7)\)

Short Answer

Expert verified
The equation of the line is \(x = -2\).

Step by step solution

01

Identify the properties of the line

Any line that is parallel to the y-axis has an undefined slope. This type of line is always in the form of a vertical line.
02

Use the point through which the line passes

Given the point (-2,7), we know that a vertical line passing through this point will have a constant x-coordinate.
03

Write the equation of the vertical line

Since the line is vertical and passes through (-2,7), the x-coordinate is always -2. Therefore, the equation of the line is \(x = -2\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

parallel to y-axis
Vertical lines are always parallel to the y-axis. This means they run up and down and never cross the y-axis. Because they don't tilt or change direction, they don鈥檛 have a slope like other lines.
A good way to look at it is that vertical lines stand 'straight up' like soldiers lined up in a row. They are set apart only by their constant x-coordinates.
Unlike horizontal lines or lines with positive or negative slopes, vertical lines are unique in their alignment parallel to the y-axis.
undefined slope
When discussing vertical lines, the term 'undefined slope' comes up a lot. But what does it mean? For most lines, the slope is a measure of how steep the line is. Mathematically, the slope is calculated by the change in y-coordinates divided by the change in x-coordinates. For vertical lines, the x-coordinates do not change.
This means you would be dividing by zero, which is mathematically undefined. That's why we say vertical lines have an undefined slope.
Thinking of it graphically, imagine a rocket shooting straight up. No matter how high it goes, the horizontal distance (or x-coordinate) stays the same.
constant x-coordinate
The concept of a constant x-coordinate is central to understanding vertical lines. A vertical line passing through a point such as \((-2, 7)\) means that at every point on the line, the x-coordinate is \'-2\'. The y-coordinate can take any value.
In general, if we know a vertical line passes through some point \'(a, b)\', the equation of the line is simply \'x = a\'. This equation tells us that x stays constant while y changes.
Using our example, the equation of the vertical line passing through the given point \(-2,7\) is \'x = -2\'. This shows the x-coordinate is always -2 regardless of the y-value.

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