Problem 59 - 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 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 line is \(y=-1/5x+6\).

Step 1: Determine the slope of the given line

In the equation \(y=5x-1\), the coefficient of \(x\) represents the slope of the line. Thus, the slope \(m1\) of the given line is 5.

Step 2: Find the slope of the perpendicular line

The slopes of two perpendicular lines are negative reciprocals of each other. Therefore, the slope \(m2\) of the line perpendicular to the given line, that we're looking to find, would be the negative reciprocal of \(m1\). So, \(m2\) would be \(-1/5\).

Step 3: Write the equation of the line

We already know the slope of the line we're finding (\(m2\)) and its y-intercept (given as 6). Substituting these values into the equation form \(y=mx+b\), where \(m\) is the slope and \(b\) is the y-coordinate of y-intercept, yields the equation of the line as \(y=-1/5x+6\).

Key Concepts

Slope of a Line

The slope of a line is a measure of its steepness and is represented by the letter 'm' in linear equations of the form y = mx + b. In more technical terms, the slope is the rate at which the variable 'y' changes for a unit change in the variable 'x'.

To calculate the slope when given two points \( (x_1, y_1) \) and \( (x_2, y_2) \), use the formula \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]. The resulting value can tell us if the line is inclining (positive slope), declining (negative slope), horizontal (zero slope), or vertical (undefined slope).

Understanding the slope is crucial when working with linear equations because it guides us in graphing lines and understanding their relationships, such as whether two lines are parallel or perpendicular.

Y-intercept

The y-intercept of a line is the point where the line crosses the y-axis. In the linear equation y = mx + b, the 'b' represents the y-intercept. This value indicates the height at which the line touches the y-axis when 'x' equals zero.

Finding the y-intercept is straightforward when you have the equation of a line. You can simply identify 'b' in the equation or set 'x' to zero and solve for 'y'. The y-intercept is particularly helpful because, along with the slope, it allows us to write the full equation of a line and understand where a line will start if plotted on a graph.

Negative Reciprocal

The concept of a negative reciprocal is essential when dealing with perpendicular lines in geometry. If two lines are perpendicular, the slope of one line is the negative reciprocal of the other. The negative reciprocal of a number is found by inverting the number (taking its reciprocal) and then changing its sign.

For instance, if one line has a slope of 'a/b', the slope of the line perpendicular to it will be '-b/a'. This relationship guarantees the perpendicularity in the context of a two-dimensional Cartesian coordinate system. It's a key concept in solving problems where you're asked to find the equation of a line that must be perpendicular to another.

Linear Equations

Linear equations are algebraic expressions that represent straight lines when graphed on a coordinate plane. The most common form of a linear equation is the slope-intercept form, \( y = mx + b \), where 'm' is the slope, and 'b' is the y-intercept.

Linear equations allow us to represent real-world and mathematical phenomena where there is a constant rate of change between variables. They are the foundation on which more complex mathematics is built. Moreover, understanding how to construct and manipulate linear equations is vital for solving a wide range of problems in algebra, such as finding the point of intersection between two lines, determining parallel or perpendicular lines, and representing various real-life situations mathematically.