Problem 39
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. $$2 x+4 y=8$$
Step-by-Step Solution
Verified Answer
The slope of a line parallel to the given line is -1/2, while the slope of a line that's perpendicular to the given line is 2.
1Step 1: Find the slope of the given line
To find the slope of the given line 2x + 4y = 8, first rearrange to the y = mx + b form where m is the slope. This yields \( y = -\frac{1}{2}x + 2 \), so the slope (\(m\)) of the given line is -1/2.
2Step 2: Find the slope of a line parallel to the given line
Parallel lines have the same slope. Therefore, the slope of a line parallel to the given line is also -1/2.
3Step 3: Find the slope of a line perpendicular to the given line
Perpendicular lines have slopes that are negative reciprocals of each other. The negative reciprocal of -1/2 is \(2\). Thus, the slope of a line perpendicular to the given line is \(2\).
Key Concepts
Slope of a LineParallel LinesPerpendicular Lines
Slope of a Line
The slope of a line is a measure of its steepness and direction. It is commonly represented by the letter "m" in the linear equation form: \( y = mx + b \), where "b" is the y-intercept. Understanding the slope is key to grasping the behavior of a linear equation.
To calculate the slope, you need to rearrange any given linear equation into the slope-intercept form (\( y = mx + b \)). This transformation helps isolate "m," making it easy to identify the slope. For example, if the equation of a line is given by \( 2x + 4y = 8 \), rearranging it yields \( y = -\frac{1}{2}x + 2 \). Here, the slope "m" is \(-\frac{1}{2}\).
Why is slope important? It tells us how much "y" changes for a change in "x." A positive slope means as "x" increases, "y" increases. A negative slope means as "x" increases, "y" decreases.
To calculate the slope, you need to rearrange any given linear equation into the slope-intercept form (\( y = mx + b \)). This transformation helps isolate "m," making it easy to identify the slope. For example, if the equation of a line is given by \( 2x + 4y = 8 \), rearranging it yields \( y = -\frac{1}{2}x + 2 \). Here, the slope "m" is \(-\frac{1}{2}\).
Why is slope important? It tells us how much "y" changes for a change in "x." A positive slope means as "x" increases, "y" increases. A negative slope means as "x" increases, "y" decreases.
- If the slope is undefined, the line is vertical.
- If the slope is zero, the line is horizontal.
Parallel Lines
Parallel lines capture a unique relationship where they never intersect. They run alongside each other indefinitely and maintain the same distance apart. For this reason, parallel lines have identical slopes. Understanding this concept is crucial when analyzing multiple lines and determining their relationships.
If you are tasked with finding the slope of a line parallel to another, simply identify the slope of the original line. For a line represented by the equation \( 2x + 4y = 8 \) or its rearranged form \( y = -\frac{1}{2}x + 2 \), the slope is \(-\frac{1}{2}\). Thus, any line parallel to this will also have a slope of \(-\frac{1}{2}\).
In essence, parallel lines in coordinate geometry are defined by precisely matching slopes:
If you are tasked with finding the slope of a line parallel to another, simply identify the slope of the original line. For a line represented by the equation \( 2x + 4y = 8 \) or its rearranged form \( y = -\frac{1}{2}x + 2 \), the slope is \(-\frac{1}{2}\). Thus, any line parallel to this will also have a slope of \(-\frac{1}{2}\).
In essence, parallel lines in coordinate geometry are defined by precisely matching slopes:
- Matching slopes result in parallel lines.
- No matter how far you extend them, they will never meet.
Perpendicular Lines
Perpendicular lines intersect at right angles (90 degrees). This special relationship involves slopes that are negative reciprocals of each other. This means, if one line has a slope "m," the line perpendicular to it will have a slope of \(-1/m\). This principle forms the cornerstone of geometry involving perpendicularity.
For instance, given a line with a slope of \(-\frac{1}{2}\), such as the line described by \( y = -\frac{1}{2}x + 2 \), the slope of any line perpendicular to it would be \(2\) (since \(-1/(-\frac{1}{2}) = 2\)).
Recognizing perpendicular lines equips you to:
For instance, given a line with a slope of \(-\frac{1}{2}\), such as the line described by \( y = -\frac{1}{2}x + 2 \), the slope of any line perpendicular to it would be \(2\) (since \(-1/(-\frac{1}{2}) = 2\)).
Recognizing perpendicular lines equips you to:
- Verify intersections forming right angles on a graph.
- Calculate precise angles between intersecting lines.
Other exercises in this chapter
Problem 38
Use intercepts and a checkpoint to graph each equation. $$y-4 x=0$$
View solution Problem 38
Determine whether each ordered pair is a solution of the given equation. $$y=4 x \quad(3,12),(12,3),(-5,-20)$$
View solution Problem 39
a. Put the equation in slope-intercept form by solving for \(y .\) b. Identify the slope and the \(y\) -intercept. c. Use the slope and y-intercept to graph the
View solution Problem 39
Use slopes to solve Exercises \(39-40\). Show that the points whose coordinates are \((-3,-3)\) \((2,-5),(5,-1),\) and \((0,1)\) are the vertices of a four-side
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