Problem 16 - Precalculus

Question

Find an equation for the line having the given slope and passing through the given point. Write your answers in the form \(y=m x+b\). (a) \(m=22 ;\) through (0,0) (b) \(m=-222 ;\) through (0,0)

Step-by-Step Solution

Answer

(a) \(y = 22x\); (b) \(y = -222x\)

Step 1: Understanding the Slope-Intercept Form

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

Step 2: Plugging in the Slope and Point (a)

For part (a), we are given \(m = 22\) and a point on the line \((0,0)\). Since the line passes through the origin (0,0), the y-intercept \(b\) is 0. Therefore, we substitute these values into the equation: \(y = 22x + 0\), simplifying to \(y = 22x\).

Step 3: Plugging in the Slope and Point (b)

For part (b), we have \(m = -222\) and the point \((0,0)\). Again, since the line passes through the origin, the y-intercept \(b\) is 0. Substitute \(m\) and \(b\) into the line equation: \(y = -222x + 0\), simplifying to \(y = -222x\).

Step 4: Confirming the Equation Format

Ensure both equations are in the format \(y = mx + b\), where the y-intercept \(b\) remains zero due to the point through which the line passes being the origin.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a common way to express the equation of a line. It is written as \(y = mx + b\), where:

• \(m\) represents the slope – a measure of the line's steepness.
• \(b\) represents the y-intercept – the point where the line crosses the y-axis.

This form is convenient because it allows you to immediately identify the slope and the y-intercept from the equation.
It is particularly useful for graphing the line or for quickly understanding how the line behaves in terms of direction and steepness.

Slope of a Line

The slope of a line, indicated by \(m\), describes how steep the line is. It tells you the direction and angle of the line. For any linear equation, the slope can be interpreted as:

• Positive Slope: When \(m > 0\), the line rises as it moves from left to right.
• Negative Slope: When \(m < 0\), the line falls as it moves from left to right.
• Zero Slope: A horizontal line where \(m = 0\).

The slope can be calculated by finding the change in \(y\) (vertical change) over the change in \(x\) (horizontal change) between two points on the line: \(m = \frac{y_2 - y_1}{x_2 - x_1}\).
This formula helps in deriving the slope when not directly given.

Y-Intercept

The y-intercept, denoted as \(b\), is the value where the line crosses the y-axis. This point occurs when \(x = 0\).

• It provides a starting point for graphing the line.
• If \(b = 0\), the line passes through the origin (0,0).

Determining the y-intercept is crucial in understanding the vertical alignment of the line on a graph.
It is also helpful when comparing different lines to see how they differ in position relative to the y-axis.

Equation of a Line

An equation of a line essentially defines the relationship between \(x\) and \(y\) for every point along that line. In the slope-intercept form, the equation can be directly formulated if you know the slope and the y-intercept.
For example, if the slope \(m = 22\) and the line crosses at \(b = 0\) (y-intercept), the equation becomes \(y = 22x\).
Lines through the origin typically have the equation in the simpler form \(y = mx\) since \(b = 0\).

• Having this equation allows you to predict \(y\) for any \(x\), graph the line, and understand its behavior.

This straightforward relationship highlights the practical use of linear equations in various applications.