Question

Find the slope and the \(y\) -intercept (if possible) of the line. $$ x=4 $$

Short answer

The slope of the line \(x = 4\) is undefined, and it does not have a y-intercept as it is a vertical line parallel to the y-axis.

Step-by-step solution

Analyzing the Given Line

The given equation is of the form \(x = c\), where \(c\) is a constant. Hence, this equation represents a vertical line at \(x=4\) on the Cartesian plane. Let's discuss its slope and y-intercept.

Determine the Slope

A vertical line does not have a defined slope. The slope of a line is defined as the change in \(y\) over the change in \(x\). For a vertical line, there's is no change in \(x\) for any non-zero variation on \(y\), which leads us to a division by zero. Therefore, the slope of this vertical line is undefined.

Determine the y-intercept

The y-intercept of a line is the point where it crosses the y-axis. However, a vertical line like \(x=4\) runs parallel to the y-axis and does not intersect it at any point. Hence, the line \(x=4\) does not have a y-intercept.

Key concepts

Undefined Slope

When we talk about the slope of a line in algebra, we're describing how steep a line is. The formula for finding the slope is \( m = \frac{\Delta y}{\Delta x} \) where \( \Delta y \) is the change in the vertical direction and \( \Delta x \) is the change in the horizontal direction. However, for a vertical line such as \( x = 4 \), there's a peculiarity: it goes straight up and down and does not incline left or right at all. As a result, for a vertical line, the change in \( x \) is always zero and since dividing by zero is undefined in mathematics, the slope for a vertical line is also said to be undefined. This is a crucial concept because it highlights a fundamental difference between vertical lines and all other types of lines which have a defined slope.

Y-Intercept

The y-intercept of a line is quite simply the point where the line crosses the y-axis on the Cartesian plane. It's where the value of \( x \) is zero. For a line given by the equation \( y = mx + b \) (where \( m \) stands for the slope and \( b \) is the y-intercept), finding this point is straightforward by looking at the value of \( b \). But what about a vertical line? Since vertical lines run parallel to the y-axis, they never actually cross it, which means that they do not have a y-intercept. This can sometimes be a tricky concept, but remembering that the y-intercept is exclusively where the line meets the y-axis should help clarify why vertical lines don't have this feature.

Cartesian Plane

The Cartesian plane is a two-dimensional surface that helps us visualize equations and coordinate points. It's composed of two perpendicular axes: the horizontal x-axis and the vertical y-axis. The intersection of these axes is known as the origin, which has coordinates \( (0,0) \). Any point on this plane can be identified by an ordered pair \( (x,y) \), representing its horizontal and vertical positions, respectively. Lines on the Cartesian plane can have various orientations and, depending on their slope, will intersect the axes at different points—except vertical lines like \( x = 4 \), which, as previously mentioned, run parallel to the y-axis and do not intersect it at all.

Equation of a Line

The equation of a line expresses the relationship between the x and y coordinates of any point on that line. The most common form of a line's equation is the slope-intercept form \( y = mx + b \), where \( m \) represents the slope and \( b \) represents the y-intercept. However, not all lines can be represented in this way. For instance, vertical lines cannot be expressed in slope-intercept form because their slope is undefined, and they lack a y-intercept. Instead, vertical lines have equations of the form \( x = c \), where \( c \) is the horizontal intercept, the x-coordinate of the point where the line crosses the x-axis. This form indicates that no matter what value y takes, x will always be \( c \)—which explains the line's vertical nature. By understanding this different form, we can more completely understand how different types of lines are represented on the Cartesian plane.