Problem 59 - Introductory Algebra for College Students

Question

Write an equation in the form \(y=m x+b\) of the line that is described. The \(y\) -intercept is 6 and the line is perpendicular to the line whose equation is \(y=5 x-1\).

Step-by-Step Solution

Answer

The equation of the desired line is \(y = -1/5x+6\).

Step 1: Understand the Characteristics of Perpendicular Lines

Two lines are perpendicular if and only if the product of their slopes is -1. Thus, if a line has a slope m, then the slope, say m', of a line which is perpendicular to it will be \(-1/m\). In this case, the slope of the given line (which our desired line is perpendicular to) is 5.

Step 2: Determine the Slope of the Desired Line

So, the slope, m, of the line we want to find (which is perpendicular to the given line) will be \(-1/5\).

Step 3: Form the Equation of the Line

Now we can form the equation of the line in the slope-intercept form \(y = mx+b\), using the determined slope, m=\(-1/5\), and the given y-intercept, b=6, which results in the equation \(y = -1/5x+6\)

Key Concepts

Slope-Intercept Form

When learning about linear equations, the slope-intercept form is one of the most common ways to express the equation of a line. It is written as \(y = mx + b\). This form is favored for its simplicity and ease of use in graphing lines.

• \(m\): This represents the slope of the line. The slope indicates the line's steepness and direction.
• \(b\): This stands for the y-intercept. It is the point where the line crosses the y-axis.

This form allows you to quickly identify both the slope and y-intercept of a line, making it straightforward to graph or analyze the line's behavior. If you know these two pieces, you can easily sketch the line or use it to solve problems related to linear relationships.

Y-Intercept

The y-intercept of a linear equation is a crucial part of understanding how the line behaves on a graph. It is represented by the point where the line crosses the y-axis, which can be directly identified from the slope-intercept form as \(b\) in the equation \(y = mx + b\).

• This is when the value of \(x\) is zero. For example, if \(b = 6\), then when \(x = 0\), \(y = 6\).
• The y-intercept provides a starting point for graphing the line, as it gives the exact location on the y-axis where the line will meet or cross.

Understanding and identifying the y-intercept is essential for constructing accurate graphs of linear equations and solving real-world problems involving intersections.

Slope of a Line

The slope of a line is a measure that describes both the angle and the direction of a line on a graph. Mathematically, the slope \(m\) is given by the ratio \(\frac{\Delta y}{\Delta x}\), which represents the change in y divided by the change in x.Key characteristics of the slope include:

• Positive Slope: Indicates that the line rises from left to right.
• Negative Slope: Reveals a falling line from left to right.
• Zero Slope: Reflects a horizontal line, meaning no change in y as x changes.
• Undefined Slope: Denotes a vertical line, where there is no change in x as y changes.

In the context of perpendicular lines, if two lines have slopes \(m_1\) and \(m_2\), they are perpendicular if \(m_1 \cdot m_2 = -1\). Therefore, understanding how to calculate and interpret the slope is vital for problems involving perpendicularity, such as deriving the equation of the line perpendicular to another.