Problem 37
Question
The equation of a line is given. Find the slope of a line that is a. parallel to the line with the given equation; and b. perpendicular to the line with the given equation. $$4 x+y=7$$
Step-by-Step Solution
Verified Answer
The slope for a line parallel to the given line is \(-4\), and for a line perpendicular to the given line is \(1/4\).
1Step 1: Convert the given line to slope-intercept form
You need to solve the equation \(4x + y = 7\) for \(y\) to put it into slope-intercept form. That will look like \(y = -4x + 7\). So, the slope (\(m\)) of the given line is \(-4\).
2Step 2: Find a line parallel to the given line
Since parallel lines have the same slope, the slope of a line parallel to the given line is also \(-4\). Take note that any line with this slope would be parallel to the given line.
3Step 3: Find a line perpendicular to the given line
The slope of the line perpendicular to a given line is the negative reciprocal of the slope of the given line. Thus, the negative reciprocal of \(-4\) which is \(-1/-4 = 1/4\). Therfore, the slope of the line perpendicular to the given line is \(1/4\).
Key Concepts
Parallel LinesPerpendicular LinesSlope-Intercept Form
Parallel Lines
When we talk about parallel lines, we mean two lines that run alongside each other at a consistent distance apart. These lines will never meet, no matter how far they are extended.
The key characteristic of parallel lines is that they have the same slope. If you imagine two roads running parallel, they're inclined at exactly the same angle. So, the equation of the line provided in the problem, which is in slope-intercept form as \[ y = -4x + 7 \]shows a slope of \(-4\).Any line that is parallel to this given line also has a slope of \(-4\).So, no matter the y-intercept, a parallel line will stay forever parallel as long as the slopes match. Remember that the y-intercept can vary, but as long as the slope is \(-4\), the lines are parallel.
The key characteristic of parallel lines is that they have the same slope. If you imagine two roads running parallel, they're inclined at exactly the same angle. So, the equation of the line provided in the problem, which is in slope-intercept form as \[ y = -4x + 7 \]shows a slope of \(-4\).Any line that is parallel to this given line also has a slope of \(-4\).So, no matter the y-intercept, a parallel line will stay forever parallel as long as the slopes match. Remember that the y-intercept can vary, but as long as the slope is \(-4\), the lines are parallel.
Perpendicular Lines
Perpendicular lines give us a completely different story. While parallel lines never intersect, perpendicular lines intersect at a right angle, meaning they meet and form a 90-degree angle.
The rule for perpendicular lines involves the negative reciprocal of the line's slope. The equation given provides a line with a slope of \(-4\).To find the slope of a line perpendicular to this, we find the negative reciprocal. The reciprocal of \(-4\)is \(-\frac{1}{4}\), and the negative reciprocal changes \(-\frac{1}{4}\) into \(\frac{1}{4}\). Therefore, any line with a slope of \(\frac{1}{4}\)would be perpendicular to the given line. This fundamental relationship ensures that perpendicular lines have slopes that multiply together to equal \(-1\).
This means that if one line's slope is \(-4\), a perpendicular line must slope at \(\frac{1}{4}\).Together, \(-4 \times \frac{1}{4} = -1\),satisfying the condition for perpendicularity.
The rule for perpendicular lines involves the negative reciprocal of the line's slope. The equation given provides a line with a slope of \(-4\).To find the slope of a line perpendicular to this, we find the negative reciprocal. The reciprocal of \(-4\)is \(-\frac{1}{4}\), and the negative reciprocal changes \(-\frac{1}{4}\) into \(\frac{1}{4}\). Therefore, any line with a slope of \(\frac{1}{4}\)would be perpendicular to the given line. This fundamental relationship ensures that perpendicular lines have slopes that multiply together to equal \(-1\).
This means that if one line's slope is \(-4\), a perpendicular line must slope at \(\frac{1}{4}\).Together, \(-4 \times \frac{1}{4} = -1\),satisfying the condition for perpendicularity.
Slope-Intercept Form
The slope-intercept form is a streamlined way of writing the equation of a line. It's easy to understand and helpful for quickly identifying key features of the line. The format of the slope-intercept form is \(y = mx + b\),where \(m\)represents the slope and \(b\)is the y-intercept.
In order to find the slope of a line, sometimes we first need to rearrange or solve an equation into this form. For the line in our exercise from the original equation \(4x + y = 7\),simplifying this into slope-intercept form gives us \(y = -4x + 7\).Here, it's clear that the slope \(m\)is \(-4\). This tells us precisely how steep the line is and in what direction it inclines. The slope-intercept form is a valuable tool for visualizing linear equations and determining the relationship between the two variables on the axes.
Remember, we use the slope-intercept form to quickly gather information about the slope and y-intercept, essential for understanding what lines involving these equations look like on a graph.
In order to find the slope of a line, sometimes we first need to rearrange or solve an equation into this form. For the line in our exercise from the original equation \(4x + y = 7\),simplifying this into slope-intercept form gives us \(y = -4x + 7\).Here, it's clear that the slope \(m\)is \(-4\). This tells us precisely how steep the line is and in what direction it inclines. The slope-intercept form is a valuable tool for visualizing linear equations and determining the relationship between the two variables on the axes.
Remember, we use the slope-intercept form to quickly gather information about the slope and y-intercept, essential for understanding what lines involving these equations look like on a graph.
Other exercises in this chapter
Problem 36
Use intercepts and a checkpoint to graph each equation. $$2 x+y=0$$
View solution Problem 36
In which quadrants do the \(x\) -coordinates and the \(y\) -coordinates have opposite signs?
View solution Problem 37
In Exercises \(27-38,\) graph each linear equation using the slope and y-intercept $$y=-\frac{5}{3} x$$
View solution Problem 37
On the same set of axes, draw lines passing through the origin with slopes \(-1,-\frac{1}{2}, 0, \frac{1}{3},\) and 2.
View solution