Problem 33 - 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 of the line is \(y = -2.25x + 3\).

Step 1: Identify the Given Information

We are given the slope of the line, which is \(-2.25\), and the \(y\)-intercept, which is \(3\).

Step 2: Apply the Slope-Intercept Form of a Line

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. Here, \(m = -2.25\) and \(b = 3\).

Step 3: Write the Equation

Substitute the given slope and \(y\)-intercept into the slope-intercept form: \(y = -2.25x + 3\). This is the equation of the line.

Key Concepts

Understanding Linear Equations

Linear equations are a fundamental concept in algebra, representing a straight line on the Cartesian coordinate plane. These equations usually take the form of \(y = mx + b\), known as the slope-intercept form. Here, each term has a specific purpose and meaning:

• \(x\) and \(y\) are variables representing any point on the line.
• \(m\) is the slope, determining how steep or flat the line is.
• \(b\) is the \(y\)-intercept, indicating where the line crosses the \(y\) axis.

By understanding these components, you can create an equation that describes the relationship between \(x\) and \(y\). This is helpful for graphing lines and solving problems involving linear relationships.

What is a Slope?

The slope of a line in a linear equation shows how the line moves across the coordinate plane. It's essentially a measure of its steepness or incline, and it's represented by \(m\) in the equation \(y = mx + b\). Here's how you can think about slope:

• If the slope is positive, the line rises from left to right.
• If the slope is negative, the line falls from left to right, as in our example where \(m = -2.25\).
• A larger absolute value of the slope indicates a steeper line.
• A slope of zero means the line is perfectly horizontal.

The slope can be calculated using the formula \[ m = \frac{\Delta y}{\Delta x} \]where \(\Delta y\) is the change in \(y\) and \(\Delta x\) is the change in \(x\). Understanding the slope helps in analyzing the direction and steepness of a line.

Exploring the Y-Intercept

The \(y\)-intercept is a crucial point on the line described by a linear equation, specifically the point where the line crosses the \(y\)-axis. In the formula \(y = mx + b\), \(b\) represents the \(y\)-intercept. Its value tells you the exact point where the line starts when moving along the horizontal axis:

• When \(x = 0\), the value of \(y\) is equal to the \(y\)-intercept \(b\).
• In our example, the \(y\)-intercept is \(3\), so the line crosses the \(y\)-axis at the point \((0, 3)\).

The \(y\)-intercept is significant because it provides a starting point for graphing the line. It's a constant that affects the vertical positioning of the line without changing its slope. Remember, changes in \(b\) will shift the entire line up or down without altering its steepness.