Problem 66 - Introductory and Intermediate Algebra for College Students 4th

Question

Write an equation in slope-intercept form of the line satisfying the given conditions. The line is perpendicular to the line whose equation is \(4 x-y=6\) and has the same \(y\) -intercept as this line.

Step-by-Step Solution

Answer

The equation of the line that is perpendicular to the line \(4x-y=6\) and has the same y-intercept is \(y=-1/4x-6\).

Step 1: Find the slope and y-intercept of the given line

First, we need to convert the equation \(4x-y=6\) into slope-intercept form, which is \(y=mx+b\), where \(m\) is the slope and \(b\) is the y-intercept. To do this, we solve for \(y\): \(y=4x-6\). In this form, we can identify \(m=4\) as the slope of the line and \(b=-6\) as the y-intercept.

Step 2: Determine the slope of the perpendicular line

Two lines are perpendicular if the product of their slopes is -1. If the slope of the given line is 4, the slope of a line perpendicular to it, denoted \(m'\), can be found using the equation \(m*m'=-1\), which gives \(m'=-1/m=-1/4\). Therefore, the slope of the line we are trying to find is \(-1/4\).

Step 3: Write the equation of the required line

Finally, we can write the equation of the required line in slope-intercept form using the slope(\(m'=-1/4\)) found in step 2 and the y-intercept (\(b=-6\)) of the given line. The result is \(y=-1/4x-6\).

Key Concepts

Slope-Intercept Form

The slope-intercept form is essential for understanding linear equations. This form of a line's equation is written as \(y = mx + b\), where \(m\) represents the slope and \(b\) represents the y-intercept.
It provides a straightforward way to graph a line and understand its basic characteristics such as steepness and where it crosses the y-axis.

• Slope \(m\): Indicates how steep the line is. A larger slope means a steeper line.
• Y-intercept \(b\): The point where the line crosses the y-axis. It is essentially the line's starting point on the graph.

Understanding this form allows you to quickly determine these crucial aspects of a line, making graphing, analyzing, and comparing lines much easier.

Slope of a Line

The slope, denoted as \(m\), indicates how steep a line is and the direction it goes. It is calculated as the "rise over run," which means how much the line goes up (or down) for each unit it moves to the right along the x-axis.

• If the slope \(m\) is positive, the line goes upwards as it moves to the right.
• If the slope \(m\) is negative, the line goes downwards as it moves to the right.
• A larger absolute value of the slope indicates a steeper line.

When two lines are perpendicular, their slopes have a special relationship: their product is -1. For example, if one line has a slope of 4, a line perpendicular to it would have a slope of \(-\frac{1}{4}\). This geometric property ensures that the lines intersect at right angles.

Y-Intercept

The y-intercept, represented as \(b\) in the slope-intercept form \(y = mx + b\), is where the line crosses the y-axis. This point is critical because it provides the graph's starting benchmark point—in this case, when \(x = 0\), \(y\) equals the y-intercept.

• The y-intercept is a fixed point on the graph, providing a reliable reference for plotting the line.
• This element of the line doesn't change unless the whole line moves up or down without changing its slope.

In our exercise, the y-intercept is \(-6\), showing that at the point where \(x = 0\), \(y = -6\). Keeping the y-intercept the same for lines being compared or manipulated, as in this exercise, helps maintain consistency in vertical placement across the graph. This consistent y-intercept shows that while the slopes adjust (as seen with perpendicular lines), they all cross the y-axis at the same point.