Problem 33 - Brief Applied Calculus

Question

Write an equation of the line satisfying the following conditions. If possible, write your answer in the form \(y=m x+b\). Slope \(-2.25\) and \(y\) -intercept 3

Step-by-Step Solution

Answer

The equation is \(y = -2.25x + 3\).

Step 1: Understand the Problem

We are given a slope of -2.25 and a y-intercept of 3. We need to write the equation of a line using these two pieces of information.

Step 2: Use the Slope-Intercept Form

The slope-intercept form of a line is given by the equation \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

Step 3: Substitute the Values

Substitute the given slope \(-2.25\) for \(m\) and the y-intercept 3 for \(b\) into the slope-intercept form equation: \(y = -2.25x + 3\).

Key Concepts

Equation of a Line

An equation of a line is a mathematical representation that describes all the points on a line. One of the most common forms used to write the equation of a line is the slope-intercept form. This form makes it straightforward to identify important characteristics of the line, like the slope and the y-intercept. In general, line equations can be used in a variety of fields such as physics, engineering, and economics to model relationships between variables.

There are different forms of linear equations, but the slope-intercept form is frequently used due to its simplicity. When writing the equation of a line, especially when given a slope and a y-intercept, the goal is to create an equation that others can use to graph or analyze the line easily. This form allows for a straightforward substitution of values, making it an efficient way to understand the properties of the line.

Slope

The slope of a line measures its steepness, which essentially tells us how much the line rises or falls as we move from one point to another along the line. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. The slope is denoted by the letter \(m\) in equations.

• If the slope is positive, the line rises from left to right.
• If the slope is negative, the line falls from left to right.
• If the slope is zero, the line is horizontal.
• If the slope is undefined, the line is vertical.

In our specific problem, the slope is given as \(-2.25\), indicating that the line is descending from left to right. The exact value of the slope tells us that for every one unit we move to the right, the line drops by 2.25 units. Knowing the slope helps in predicting and understanding the direction and angle of the line, which is crucial in various analytical applications.

Y-intercept

The y-intercept of a line is the point where the line crosses the y-axis. This is the value of \(y\) when \(x\) is zero, and it is represented by \(b\) in the slope-intercept form \(y = mx + b\). The y-intercept provides a starting point on the graph and is essential for plotting the rest of the line.

For instance, in this exercise, the y-intercept is 3, meaning the line crosses the y-axis at the point (0, 3).

The y-intercept allows for easy predictions about the behavior of the line at the boundaries of a graph, especially when combined with the slope. In real life scenarios, the y-intercept can represent initial conditions or starting values in various applications like budgeting, physics experiments, or any scenario requiring a projection from a certain point.