Problem 60 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(57-64\), write an equation in the form \(y=m x+b\) of the line that is described. The \(y\) -intercept is 7 and the line is perpendicular to the line whose equation is \(y=8 x-3\)

Step-by-Step Solution

Answer

The equation of a line that is perpendicular to the given line \(y=8x-3\) with a y-intercept of 7 is \(y=-1/8x + 7\).

Step 1: Identify the slope of the given line

The equation \(y=8x-3\) is in slope-intercept form, therefore its slope (m) is 8.

Step 2: Determine the slope of the line that is perpendicular to the given line

Since the slope m1 of the given line is 8, the slope m2 of a line perpendicular to it will satisfy the equation \(m1 *m2 =-1\). So, it will be \(-1/8 \).

Step 3: Write the equation of the line

Using the slope of -1/8 found in the previous step and the given y-intercept (b) of 7, you can write the equation of the line in slope-intercept form (y=mx+b) as- \(y= -1/8x + 7\)

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is one of the simplest ways to describe a straight line on a graph. This form is expressed as \( y = mx + b \), where:

• \( y \) is the dependent variable (usually the vertical axis on a graph)
• \( x \) is the independent variable (typically the horizontal axis)
• \( m \) represents the slope of the line, which measures how steep the line is
• \( b \) is the y-intercept, which indicates where the line crosses the y-axis

This form is highly useful because it immediately tells you both the slope and the y-intercept, which are crucial for understanding the behavior and position of a line. Additionally, using the slope and y-intercept makes it easy to graph the line. Simply start at the y-intercept \( b \) on the y-axis, and from this point, use the slope \( m \) to determine another point. For example, a slope of \( 2 \) means you go up two units for every unit you move to the right.

Perpendicular Lines

Perpendicular lines are lines that intersect at a right angle (90 degrees) to each other. In terms of their slopes, the relationship between two perpendicular lines can be expressed using their slopes, \( m_1 \) and \( m_2 \). Lines are perpendicular if the product of their slopes is \(-1\), which can be understood as:

• \( m_1 \times m_2 = -1 \)
• This means if one line has a slope \( m_1 \), the other line will have a slope that is the negative reciprocal of \( m_1 \)

For example, if a line has a slope of \( 8 \), a line perpendicular to it will have a slope of \(-\frac{1}{8}\), as demonstrated with the exercise involving the equation \( y = 8x - 3 \). Understanding this concept is crucial when you need to create or find equations of lines that need to be perpendicular to each other.

Y-Intercept

The y-intercept is one of the most important aspects to consider when examining the slope-intercept form of a line, denoted by \( b \) in the equation \( y = mx + b \). It represents the point on the graph where the line crosses the y-axis. This means:

• At the y-intercept, the value of \( x \) is always zero
• The y-intercept provides a starting point for graphing a line when you also know the slope \( m \)

For instance, in the exercise you encountered, the y-intercept is \( 7 \). This means that the line \( y = -\frac{1}{8}x + 7 \) will cross the y-axis at \( (0, 7) \). Knowing this value helps quickly sketch how the line behaves relative to both axes and is particularly useful for providing context to linear relationships or understanding initial conditions in word problems.