Problem 52 - Precalculus Mathematics for Calculus

Question

Find the slope and \(y\) -intercept of the line and draw its graph. $$x=-5$$

Step-by-Step Solution

Answer

Slope is undefined, no y-intercept, and the line is vertical at \(x=-5\).

Step 1: Interpret the Equation

The equation given is in the form of a vertical line, which is represented by an equation of the form \(x = c\) where \(c\) is a constant. In this case, \(x = -5\).

Step 2: Understand the Slope

The slope of a line is defined as the change in \(y\) over the change in \(x\). For vertical lines, \(x\) is constant, meaning there is no change in \(x\). The slope is therefore undefined.

Step 3: Determine the Y-Intercept

The \(y\)-intercept is the point where the line crosses the \(y\)-axis. Since vertical lines do not cross the \(y\)-axis, there is no \(y\)-intercept for a vertical line like \(x = -5\).

Step 4: Visualize the Graph

Draw the graph by plotting a vertical line at \(x = -5\). This line will be parallel to the \(y\)-axis and can run infinitely in both the upward and downward directions.

Key Concepts

Slope of a Line

The slope of a line is a measure of how steep or flat a line is. It indicates the rate of change between two variables, usually represented as the vertical change (rise) over the horizontal change (run). Mathematically, the slope \(m\) is given by the formula:\[m = \frac{\Delta y}{\Delta x}\]where \(\Delta y\) is the change in \(y\) values, and \(\Delta x\) is the change in \(x\) values.
\[\]For most lines, the slope is a finite number, but for special lines like vertical lines, the situation is different. In the case of a vertical line represented by \(x = -5\), the change in \(x\) is zero because \(x\) does not vary. Since division by zero is undefined, the slope of vertical lines is considered undefined. This means the line can't be expressed in terms of how one variable changes with respect to the other, emphasizing its vertical nature.

Y-Intercept

The \(y\)-intercept is the point where a line crosses the \(y\)-axis. This point is crucial for understanding where a line begins to intersect with the vertical axis on a graph.
\[\]For most lines, the \(y\)-intercept is found by substituting 0 in place of \(x\) in the line's equation and solving for \(y\). The general form of a linear equation is \(y = mx + b\), where \(b\) represents the \(y\)-intercept.
\[\]However, for vertical lines such as \(x = -5\), there is no \(y\)-intercept. Since vertical lines run parallel to the \(y\)-axis and have no intersection point, this highlights their unique positioning compared to typical linear graphs.

Graphing Linear Equations

Graphing linear equations helps visualize how lines behave and provides insight into their properties, such as slope and intercepts. When approaching graphing, it's essential to:

• Identify the type of line: is it horizontal, vertical, or slanted?
• Determine the slope from the equation if applicable.
• Find the \(y\)-intercept, unless dealing with a vertical line.

In the example \(x = -5\), the graphing process leads to plotting a vertical line. This vertical line runs parallel to the \(y\)-axis, directly through \(x = -5\) on the \(x\)-axis.
\[\]Although vertical lines might initially seem challenging due to their undefined slope and lack of \(y\)-intercepts, plotting them is straightforward. Just draw a straight line through the specified \(x\)-coordinate extending infinitely up and down, making sure it doesn't cross the \(y\)-axis.