Problem 3 - Precalculus

Question

Compute the slope of the line passing through the two given points. In Exercise \(3,\) include a sketch with your answers. (a) (1,1),(-1,-1) (b) (0,5),(-8,5)

Step-by-Step Solution

Answer

(a) Slope = 1; the line is diagonal. (b) Slope = 0; the line is horizontal.

Step 1: Understanding the Formula

The slope of a line passing through two points \((x_1, y_1)\) and \(x_2, y_2)\) is given by the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). We'll use this formula to calculate the slope for each part of the exercise.

Step 2: Calculate Slope for Part (a)

For the points \( (1,1) \) and \( (-1,-1) \), we apply the formula: \[ m = \frac{-1 - 1}{-1 - 1} = \frac{-2}{-2} = 1. \] The slope of the line through these points is 1.

Step 3: Calculate Slope for Part (b)

For the points \( (0,5) \) and \( (-8,5) \), we use the formula: \[ m = \frac{5 - 5}{-8 - 0} = \frac{0}{-8} = 0. \] The slope of the line through these points is 0.

Step 4: Sketch the Lines

For part (a), sketch a diagonal line through \( (1,1) \) and \( (-1,-1) \) with a positive slope ascending from left to right. For part (b), sketch a horizontal line through \( (0,5) \) and \( (-8,5) \); the line is flat indicating a slope of 0.

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, bridges algebra and geometry using points on the coordinate plane to solve geometric problems. This system relies on the Cartesian coordinates, which use two number lines, one horizontal (x-axis) and one vertical (y-axis), intersecting at the origin.
With coordinate geometry, you can:

• Identify the position of a point using an ordered pair \(x, y\).
• Calculate the distance between two points with the distance formula.
• Determine the slope of a line that connects two points.
• Understand how lines and curves interact on a graph.

The benefit of using this structured system is that it provides a clear numerical way to describe geometric concepts visually. This allows anyone working in science or mathematics to more easily solve a variety of problems.

Linear Equations

Linear equations are mathematical expressions that model relationships based on straight lines. In the simplest form, a linear equation is written as:\[ y = mx + c \]where \(m\) represents the slope, and \(c\) is the y-intercept. The slope measures the steepness or angle of the line, while the intercept indicates where the line crosses the y-axis.
To find the slope of a line going through two points, such as in our exercise, you can use:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]This formula comes from the idea of "rise over run" – how much the line goes up (rise) for each step it goes over (run). A positive slope means the line ascends, while a negative slope descends.
Knowing how to work with linear equations is vital in various fields, from predicting trends in data to designing systems in engineering.

Mathematical Sketching

Mathematical sketching involves drawing representations of mathematical concepts, like lines or curves, on a graph. This helps in visualizing the relationships described by the equations.
When sketching a line:

• Start by plotting the points given.
• Draw a line through these points based on their calculated slope.
• For a slope of 1, as in part (a), your line will be at a 45-degree angle from the axis.
• For a slope of 0, as in part (b), draw a straight horizontal line.

Sketches are essential for understanding mathematical issues because they provide a clear and immediate visual representation. They help in predicting the behavior of mathematical models and solving problems accurately.