Problem 3 - Technical Mathematics with Calculus

Question

Find the slope of each straight line. Rise \(=-4.25,\) run \(=5.33\)

Step-by-Step Solution

Answer

The slope of the line is approximately -0.797.

Step 1: Understanding Slope

The slope of a straight line can be found using the formula slope (m) = rise/run. The 'rise' refers to the vertical change between two points on a line, while 'run' refers to the horizontal change between the same two points.

Step 2: Substitute Values for Rise and Run

Here, the rise is given as -4.25, and the run is given as 5.33. To find the slope, substitute these values into the slope formula: \(m = \frac{rise}{run}\).

Step 3: Calculate the Slope

Substituting the given values into the slope formula gives us \(m = \frac{-4.25}{5.33}\). Now we need to divide -4.25 by 5.33 to get the numerical value of the slope.

Step 4: Final Result

After dividing -4.25 by 5.33, we get the slope of the line: \(m = -0.797\). This is the slope of the line with the given rise and run.

Key Concepts

Slope Formula

Understanding how to calculate the slope of a line is crucial when working with linear equations in algebra. The slope, symbolized by the letter m, quantifies the steepness and direction of a line. The formula to determine the slope between any two points on the line is quite simple:
\[ m = \frac{\text{rise}}{\text{run}} \]
The numerator, or 'rise', measures the vertical change between two points on the line. On the other hand, the denominator, 'run', measures the horizontal distance. If the rise is positive, the line goes upwards as it moves from left to right. If it's negative, the line goes downwards. Similarly, the sign of the slope can indicate whether the line is increasing or decreasing.

Vertical Change (Rise)

Defining the Rise

When looking at any straight line on a graph, the rise is the change in elevation from one point to another along the y-axis, or the vertical axis. It represents how much the line goes up or down. To measure the rise, pick two points on the line, and simply subtract the y-coordinate of the first point from the y-coordinate of the second point. If the result is positive, the line rises as it moves along; if negative, it falls.

Effect of Rise on the Slope

A larger absolute value of the rise indicates a steeper line. The rise is a key part in determining the slope and, consequently, the behavior of the line on the graph.

Horizontal Change (Run)

Defining the Run

Parallel to the concept of rise, the run quantifies the horizontal movement between two points on a line along the x-axis. To determine the run, subtract the x-coordinate of the first point from the x-coordinate of the second point. The run can be thought of as the horizontal baseline over which the rise 'climbs' or 'descends' to form the slope of the line.

Role of Run in Slope Calculation

As part of the slope formula, the run is critical in balancing the slope's value. A larger run compared to the rise will result in a flatter slope, while a smaller run relative to the rise gives a steeper slope. Zero run indicates a vertical line, which is an important exception because vertical lines do not have a defined slope.

Linear Equations

Linear equations form a fundamental part of algebra and graphically represent straight lines on the Cartesian plane. Such equations typically take the form
\[ y = mx + b \]
where m represents the slope and b is the y-intercept, or the point where the line crosses the y-axis. Every linear equation exhibits a constant rate of change, which we understand as the slope. The ability to find the slope from two points on a line empowers you to write the equation of a line, predict values within the system, and understand the relationship between variables. Recognizing that linear equations model real-world situations, mastering their properties, including the calculation of slope, is incredibly useful for interpreting and predicting linear patterns in data.