Problem 4 - Technical Mathematics with Calculus

Question

Write the equation of each straight line in slope-intercept form, and make a graph. Slope \(=-2 ; y\) intercept \(=3\)

Step-by-Step Solution

Answer

The equation of the line in slope-intercept form is y = -2x + 3.

Step 1: Identify Slope and y-intercept

From the given information, the slope (m) is -2, and the y-intercept (b) is 3. Slope-intercept form of a line is given by the equation y = mx + b.

Step 2: Write the Equation Using Slope-Intercept Form

Substitute the slope and y-intercept into the slope-intercept equation: y = mx + b. Here, m = -2 and b = 3, so the equation becomes y = -2x + 3.

Step 3: Plot the y-intercept on the Graph

First plot the y-intercept on the graph. This is the point (0, 3) where the line crosses the y-axis.

Step 4: Use the Slope to Plot Another Point

Starting from the y-intercept, use the slope to find another point. Since the slope is -2, it means we need to go down 2 units and move 1 unit to the right from (0, 3). This gives us the point (1, 1).

Step 5: Draw the Line

Draw a line through the two points, (0, 3) and (1, 1). This is the graph of the equation y = -2x + 3.

Key Concepts

Linear Equations

Linear equations form the foundation for understanding how to characterize relationships between two variables in a straight line. These equations are algebraic expressions that equate to a constant value, typically in the two-dimensional coordinate plane.

They come in various formats, but one of the most common and useful is the slope-intercept form, which can be written as y = mx + b. In this form, m represents the slope of the line, which is the rate of change of the dependent variable, y, with respect to the independent variable, x. The term b signifies the y-intercept of the line, which is the point where the line crosses the y-axis.

The slope-intercept form is especially helpful since it directly shows the key characteristics of the line. Understanding how to manipulate and solve linear equations in this form is crucial for a variety of mathematical concepts and real-world applications.

Graphing Lines

Graphing lines on a coordinate plane is a visual way of representing linear equations and demonstrating the correlation between two variables. When we graph a line, we're depicting all the points that satiate the equation of the line.

To graph a line using the slope-intercept form, follow these simple steps: First, identify the y-intercept (b) on the graph and plot that point. It's where the line will cross the y-axis. Next, use the slope (m) to determine another point on the line. If the slope is positive, from the y-intercept, move upward and to the right. If the slope is negative, move downward and to the right (or upward and to the left). Finally, draw a straight line through the plotted points to complete the graph of the line.

This process illustrates how the abstract slope-intercept equation translates into a visual representation of a straight line, which helps immensely in understanding concepts such as rate of change and intercepts in real-life scenarios as well.

Slope and Y-Intercept

The slope and y-intercept are significant components of linear equations and their graphs. The slope, denoted as m in the slope-intercept formula, tells us how steep the line is, as well as the direction in which the line runs. A positive slope indicates the line rises from left to right, while a negative slope indicates it falls.

The y-intercept, represented by b, provides a starting point for the line on the graph. It is the coordinate at which the line touches the y-axis, indicated as (0, b). Knowing the y-intercept allows us to begin the graphing process at an exact location.

Together, the slope and y-intercept enable us to graph a line without requiring numerous points. For example, if we have a slope of -2 and a y-intercept of 3, we start plotting at (0, 3) and then move according to our slope—down 2 units for every 1 unit we move to the right—to get our next point. By connecting these points, we create a precise representation of the given linear equation.