Problem 34 - College Algebra and Calculus: An Applied Approach

Question

Find an equation of the line that passes through the point and has the indicated slope. Then sketch the line. Point \(\quad\) Slope \(\begin{array}{ll}\underline{\phantom{xxx}}(3,-2) & m \text { is undefined. }\end{array}\)

Step-by-Step Solution

Answer

The equation of the line is \(x = 3\).

Step 1: Identify the given point

The given point is (3, -2). Therefore, this vertical line is defined at the x-coordinate 3.

Step 2: Construct the Line Equation with Undefined Slope

When dealing with an undefined slope, the line will be vertical passing through the x-coordinate of the given point. Hence, the equation of the line will be \(x = a\), where 'a' is the x-coordinate of the given point.

Step 3: Substitute x-coordinate into Equation

Substitute x-coordinate 3 from the given point (3, -2) into the equation. The equation of the line becomes \(x = 3\).

Step 4: Sketching the Line

Draw a vertical line through the x-coordinate at 3 marking the point (3, -2). This line will pass through this point and other points such as (3,1), (3,0), (3,-1), etc. This vertical line is the graphical representation of our line equation \(x = 3\).

Key Concepts

Undefined Slope

When talking about slopes, the term "undefined slope" can be a bit tricky. In algebra and geometry, a slope defines the steepness and direction of a line. Slopes are usually represented by the letter \(m\).

Let's break it down:

• An undefined slope occurs when a line is vertical.
• Imagine trying to walk straight up a wall; you can't, thus it feels like an infinite climb, resulting in an undefined slope.
• Mathematically, a slope is undefined because the change in the x-values is zero, making the denominator of the slope formula zero, leading to division by zero.

In our case, the equation \(x = 3\) forms a vertical line, hence the slope is undefined.

Vertical Line

A vertical line is an interesting topic in geometry and algebra. Here's why:

• It runs straight up and down on the graph paper, cutting through the x-axis at a single point.
• This line doesn’t tilt left or right, remaining perfectly vertical.
• Its primary feature is that it has an undefined slope—no tilt, just a straight-up position.

For example, in our solved problem, the equation \(x = 3\) represents a vertical line. It exists at \(x = 3\), but it touches the y-axis at every possible value.

Line Equation

Formulating a line equation involves understanding how points and slopes relate. For most lines, the slope-intercept form \(y = mx + b\) is used. But for vertical lines, it's slightly different:

• The equation relates to just the x-coordinate, which remains constant, creating a line \(x = a\).
• For the given example, \(x = 3\) shows that the x-coordinate is unchanging at 3 no matter what the y-value is.
• This line equation simply states that every point along the line will have an \(x\)-value of 3.

Coordinate Geometry

Coordinate geometry is a fantastic bridge between algebra and geometry. It uses a coordinate plane to visualize algebraic equations. Here are some highlights:

• The plane is divided by the x and y axes, used to describe locations with points like \((3, -2)\).
• Coordinates show exactly where a point sits on this graph.
• In our problem, the point \((3, -2)\) dictates where our line should pass through.

Understanding these relationships helps to grasp how equations translate into graphical shapes, like our vertical line resulting from the equation \(x = 3\). It continually crosses the point \((3, -2)\) and any point with \(x\) as 3.