Problem 21 - Precalculus with Limits

Question

In Exercises 17-28, find the slope and \(y\)-intercept (if possible) of the equation of the line. Sketch the line. \( 5x - 2 = 0 \)

Step-by-Step Solution

Answer

This given line equation represents a vertical line that intersects the x-axis at \(x = 0.4\). It does not have a slope in the traditional sense and it does not intersect the y-axis. Hence, it does not have a traditional y-intercept.

Step 1: Identify the line equation form

The equation \(5x - 2 = 0\) is given in the form of a vertical line equation \(x = a\), after rearranging it becomes \(x = 0.4\).

Step 2: Identify the slope and y-intercept

The line is vertical, so it does not have a slope in the traditional sense, since the 'run' part of 'rise over run' is zero (a vertical line runs straight up and down). Additionally, a vertical line intersects the y-axis only if \(a = 0\), i.e., the constant part of the equation (\(x = a\)). However, in this case, \(a = 0.4\), which means the line does not intersect the y-axis, and hence does not have a y-intercept.

Step 3: Sketch the line

The line can be sketched as a vertical line passing through the x-axis at \(x = 0.4\).

Key Concepts

Linear Equations

Understanding linear equations is fundamental in mathematics, as they are used to describe a relationship between two variables in a straight line. A linear equation can be written in the form \( y = mx + b \) where \( m \) represents the slope, and \( b \) represents the y-intercept.

The slope, \( m \), indicates how steep the line is and the direction it travels. Positive slope means the line rises from left to right, a negative slope means it falls, and a slope of zero means the line is horizontal. The y-intercept, \( b \) is the point where the line crosses the y-axis. This is the value of \( y \) when \( x = 0 \).

In our exercise, the equation \( 5x - 2 = 0 \) doesn't fit the standard form directly because it only has an \( x \) component. If we were to write it in a more familiar format, it would not have a y term, which is atypical for linear equations we use to graph a line with a slope and intercept.

Graph Sketching

Graph sketching is a valuable skill that assists in the visualization of equations. To sketch a graph of a linear equation, you usually find two key components: the slope and y-intercept. Once you have those, you can plot the y-intercept on the y-axis, and then use the slope to determine the direction and steepness of the line.

However, there are special cases, such as the vertical line in our exercise, where traditional sketching techniques don't apply. Here, you cannot use slope-intercept form, but instead, you identify the x-value where the line will pass through and draw a straight line parallel to the y-axis at that x-value. This is a distinctively different approach compared to typical line graphing.

Vertical Line Equation

The equation of a vertical line is unique in that it takes the form \( x = a \), where \( a \) is a constant. This signifies that for all points along this line, the \( x \) value remains the same while the \( y \) value can be any number. Unlike most linear equations, a vertical line has an undefined slope because the change in \( y \) is not relative to a change in \( x \); instead, \( x \) is constant.

For vertical lines, there's no y-intercept since the concept of the y-intercept assumes that the line crosses the y-axis, which doesn't happen unless \( a = 0 \). As such, with our exercise's equation \( x = 0.4 \) and considering the definition of a vertical line, the line does not intersect the y-axis and thus has no y-intercept to speak of. Graphing this is a matter of drawing a straight line along \( x = 0.4 \) from the bottom of the graph to the top, reflecting all possible \( y \) values for this \( x \) coordinate.