Question

Slope Find the slope of each straight line. Rise \(=6 ;\) run \(=4\)

Short answer

The slope of the line is \(\frac{3}{2}\).

Step-by-step solution

Understand the Concept of Slope

The slope of a straight line is a measure of its steepness and is calculated by the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. The formula for the slope, often represented as 'm', is given by the equation \( m = \frac{\text{rise}}{\text{run}} \).

Apply the Slope Formula

Using the given values for rise and run, plug them into the slope formula. Thus, \( m = \frac{6}{4} \).

Simplify the Fraction

Simplify the fraction \( \frac{6}{4} \) by dividing the numerator and the denominator by their greatest common divisor (GCD), which is 2. So \( m = \frac{6 \div 2}{4 \div 2} = \frac{3}{2} \).

Key concepts

Slope Formula

Understanding the slope formula is essential when working with linear equations and graphing lines. The slope, often denoted as 'm', can be thought of as the direction and the steepness of a line. The formula to calculate the slope between two points on a line is quite simple:
\[\begin{equation}m = \frac{{rise}}{{run}}\end{equation}\]
Where the 'rise' refers to the vertical change and the 'run' refers to the horizontal change as you move from one point to another along the line. In essence, it's a measure of how much the line goes up or down as you travel along it. This formula is the cornerstone for many areas of algebra and calculus, as it is a fundamental aspect of understanding how graphs represent equations and how those equations describe the world.

Rise Over Run

The terminology 'rise over run' is a colloquial way of articulating the components of the slope formula. It describes literally what happens when you try to move along the line: how much you have to 'rise' (go up or down) and how much you 'run' (go left or right).

Visualizing Rise Over Run

Imagine you are walking up a hill: the 'rise' is the elevation you gain with each step forward, which is your 'run'. This visual can help students better grasp the concept as they think about the slope of a hill they might have climbed.

Calculation Example

In the given exercise, a line that rises 6 units for every 4 units it runs horizontally has a slope calculated as follows:
\[\begin{equation}m = \frac{{rise}}{{run}} = \frac{{6}}{{4}}\end{equation}\]
This formula is vital for graphing a line and understanding its behavior on a coordinate plane.

Simplifying Fractions

Simplifying fractions is a critical step in mathematics to make numbers more manageable and to reveal the most simplified form of a ratio. Fractions are simplified by dividing both the numerator (top number) and the denominator (bottom number) by their greatest common divisor (GCD).

Finding the GCD

To find the GCD of two numbers, you can list out the factors of each number and find the highest number that appears in both lists. Alternatively, use the Euclidean algorithm for a more efficient approach, especially with larger numbers.

Simplification Process

Once you've found the GCD, divide both the numerator and the denominator by it, as seen in the exercise:
\[\begin{equation}\frac{{6}}{{4}} = \frac{{6 \div 2}}{{4 \div 2}} = \frac{{3}}{{2}}\end{equation}\]
The result is a simplified fraction that represents the same value as the original, but in its simplest form.

Steepness of a Line

The steepness of a line is visually and mathematically determined by its slope. A line with a higher slope value is steeper, which translates to a greater vertical change for a given horizontal distance. Conversely, a smaller slope value indicates a less steep line.

Comparing Steepness

For example, a slope of \( \frac{3}{2} \) is steeper than a slope of \( \frac{1}{2} \) because with each step horizontally, the rise for the first slope is greater.
The concept of steepness helps contextualize the meaning of the slope: steeper lines on a graph indicate rapid changes, while less steep lines show more gradual trends. This idea is utilized in various disciplines, including physics for velocity graphs and in economics for cost curves.
Understanding the physical representation of slope as steepness aids students in visualizing and interpreting linear relationships more effectively.