Problem 20 - Contemporary Precalculus

Question

Find the equation of the line with y-intercept b and slope \(m\). $$b=-4.5, m=2.5$$

Step-by-Step Solution

Answer

Answer: The equation of the line is y = 2.5x - 4.5.

Step 1: Write the slope-intercept form of a linear equation

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

Step 2: Plug in the slope and y-intercept

We are given the slope \(m = 2.5\) and the y-intercept \(b = -4.5\). We plug these values into the slope-intercept form: $$y = 2.5x - 4.5$$

Step 3: Write the final equation of the line

With the slope and y-intercept plugged in, the equation of the line is: $$y = 2.5x - 4.5$$ This is the final equation of the line with slope \(m = 2.5\) and y-intercept \(b = -4.5\).

Key Concepts

Slope-Intercept Form

The slope-intercept form is a common way to express the equation of a straight line. In algebra, it is represented as \(y = mx + b\). Here, the equation gives a direct view into two critical parts of a line on the Cartesian plane:

• Slope \(m\): This tells us how steep the line is.
• Y-intercept \(b\): This indicates where the line crosses the y-axis.

This format makes it easy to quickly identify both the slope and y-intercept right from the equation. Hence, the name "slope-intercept form." Learning to use and manipulate this form allows you to graph lines, understand their direction, and even find parallel or perpendicular lines.

Slope

The slope of a line, denoted by \(m\), is a measure of its steepness. It shows how much the line rises vertically for a given horizontal move. In mathematical terms, slope is calculated as the change in \(y\)-value over the change in \(x\)-value, often expressed as:\[m = \frac{\Delta y}{\Delta x}\]When the slope is positive, the line inclines upwards as you move from left to right. A negative slope indicates a line that slopes downwards. A slope of zero suggests a perfectly horizontal line, while an undefined slope corresponds to a vertical line.

• A slope of 2.5, as in our example, means for every unit we move to the right along the x-axis, the line rises by 2.5 units.

Y-Intercept

The y-intercept \(b\) is where the line meets the y-axis. This is the point where \(x = 0\), and it gives crucial insight into the position of the line in relation to the y-axis. In the slope-intercept equation \(y = mx + b\), \(b\) clearly signals the starting point on the y-axis:

• If \(b\) is positive, the line crosses above the origin.
• If \(b\) is negative, it crosses below the origin. In our example, \(b = -4.5\), which means the line crosses the y-axis at -4.5.

Knowing the y-intercept allows us to accurately plot and begin graphing the line. It is a crucial part of understanding the relationship between the line and the Cartesian plane.