Problem 14 - Understanding Elementary Algebra with Geometry

Question

Find the slope of the line passing through the given points. Round to the nearest hundredth where necessary. \((3,-1)\) and \((3,2)\)

Step-by-Step Solution

Answer

The slope is undefined.

Step 1: Identify the coordinates

The coordinates given are \( (3, -1) \) and \( (3, 2). \) Let \( (x_1, y_1) = (3, -1) \) and \( (x_2, y_2) = (3, 2). \)

Step 2: Apply the slope formula

The formula for the slope \( m \) of a line passing through the points \( (x_1, y_1) \) and \( (x_2, y_2) \) is \[ m = \frac{y_2 - y_1}{x_2 - x_1}. \]

Step 3: Substitute the coordinates into the formula

Substitute the given points into the formula: \[ m = \frac{2 - (-1)}{3 - 3}. \]

Step 4: Simplify the expression

Simplify the numerator and the denominator: \[ m = \frac{2 + 1}{0} = \frac{3}{0}. \]

Step 5: Identify an undefined slope

Since the denominator is \( 0 \), the slope is undefined (or the line is vertical).

Key Concepts

coordinates

Understanding coordinates is a fundamental step in finding the slope of a line. Coordinates are written in pairs \(x, y\) and represent points on a graph. The first number in the pair is the x-coordinate, and the second number is the y-coordinate. For example, \(3, -1\) means moving 3 units to the right along the x-axis and 1 unit down along the y-axis. Remember that you always reference the x-axis first, then the y-axis. This helps you accurately identify and plot points on a graph.
In the provided exercise, we have two specific points: \(3, -1\) and \(3, 2\). Both points share the same x-coordinate, which is crucial in this problem.

slope formula

The slope formula is a crucial tool for determining the steepness and direction of a line. The formula for calculating the slope \(m\) is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \] where \(x_1, y_1\) and \(x_2, y_2\) are the coordinates of two points on the line.
This formula essentially tells us the rate of change in the y-values divided by the rate of change in the x-values.

• Positive slope: The line rises as you move from left to right.
• Negative slope: The line falls as you move from left to right.
• Zero slope: The line is horizontal.
• Undefined slope: The line is vertical.

In our example, substituting the given values:
\[ m = \frac{2 - (-1)}{3 - 3} \] simplifies to
\[ m = \frac{3}{0} \]. Ultimately, because dividing by zero is undefined, the slope here is undefined, indicating a vertical line.

undefined slope

An undefined slope occurs when the line is vertical, meaning it goes straight up and down. In the slope formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \), the denominator (\( x_2 - x_1 \)) becomes zero. Since division by zero is mathematically undefined, the slope itself is labeled as 'undefined'.
For instance, in our exercise, the points given are \(3, -1\) and \(3, 2\). Both points have the same x-coordinate. Therefore, the calculation looks like this:
\[ m = \frac{2 - (-1)}{3 - 3} = \frac{3}{0} \. \] Here, the denominator is zero, revealing an undefined slope. This tells us the line does not tilt left or right but rather runs vertically.

vertical line

A vertical line is a line where all points have the same x-coordinate. It looks like a straight line that goes up and down without tilting to either side. If you're ever uncertain whether a line is vertical, check the x-coordinates of two points on the line. If they're the same, the line is vertical.
For a vertical line:

• The slope is always undefined because you can't divide by zero.
• The line will always look the same regardless of its position on the graph, going straight up and down.
• It's often represented in equations as \( x = \text{constant} \), where 'constant' is the x-coordinate of every point on the line.

In our exercise, since both points, \(3, -1\) and \(3, 2\), share the same x-coordinate of \ 3 \, this line is indeed vertical with an undefined slope and would be represented by x = 3 in an equation.