Get started for free
Log In Start studying!
Get started for free Log out
Chapter 1: Problem 77
Find \(k\) so that the line containing the points \((-3, k)\) and \((4,8)\) isparallel to the line containing the points \((5,3)\) and \((1,-6)\)
Short Answer
Expert verified
k = \( \frac{{95}}{{4}} \).
Step by step solution
01
- Find the slope of the line containing points (5, 3) and (1, -6)
To find the slope \(m\frac{{2}}\frac{{1}}\) of a line between two points \( (x_1, y_1) \) and \( (x_2, y_2) \, use the formula \(\frac{{y_2 - y_1}}{{x_2 - x_1}}\). Here, the points are (5, 3) and (1, -6), so \(m = \frac{{-6 - 3}}{{1 - 5}} = \frac{{-9}}{{-4}} = -\frac{{9}}{{4}}s -1\right)}}{\text{4}\right)}}\frac{{6}\frac{{8}\frac{{8}}\frac{{10\ - 14}}}}}}}}\right)}}\rightarrow-\4}}}}\rightarrow.\frac{{3}\frac{{y_2}}} }right}}}}}}}}}}}}}}}}}}}}}}}})}}right}}}}0\)}}}\)}}left-dose}}}}}}}}}}}}}}}}}}}!righ}}}}}}}}}}}}\frac{{68}}-\9\ources}}}}}}}}}}}\displaystyle`}\rightarrow2}}}}}}}
02
- Write down the slope of the line containing points (-3, k) and (4, 8)
Let the slope of this line be m as well. Using the slope formula \(m=\frac{{8 - k}}{{4 - (-3)}}\), we write \(m=\frac{{8 - k}}{{7}}\).
03
- Set the slopes equal (since the lines are parallel)
Since the lines are parallel, their slopes are equal. Therefore, \( \frac{{8 - k}}{{7}} = -\frac{{9}}{{4}} \).
04
- Solve for k
Cross-multiply to solve for k:\(4(8 - k) = -9(7) \rightarrow 32 - 4k = -63 \rightarrow -4k = -63 - 32 \rightarrow -4k = -95 \rightarrow k = \frac{{95}}{{4}}\)
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Slope Formula
Finding the slope of a line is a crucial step in understanding linear equations and graphs. The slope indicates the steepness and direction of a line.
The formula for finding the slope between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is: \[ m = \frac{{y_2 - y_1}}{{x_2 - x_1}} \]
Where:
- \( m \) is the slope
- \( (x_1, y_1) \) are the coordinates of the first point
- \( (x_2, y_2) \) are the coordinates of the second point
For instance, in the example given:
- First point: \( (5, 3) \)
- Second point: \( (1, -6) \)
The slope \( m \) can be calculated as:
\[ m = \frac{{-6 - 3}}{{1 - 5}} = \frac{{-9}}{{-4}} = \frac{{9}}{{4}} \]
This slope calculation helps us understand how much the line rises or falls for each unit it moves horizontally.
Parallel Lines
Lines that are parallel have a very special property: they never intersect. This is because they have the same slope.
In our example, we need to ensure that the slope of the line passing through \( (-3, k) \) and \( (4, 8) \) is equal to the slope of the line passing through \( (5, 3) \) and \( (1, -6) \).
Let's find this second slope. Using the slope formula:
\[ m = \frac{{8 - k}}{{4 - (-3)}} = \frac{{8 - k}}{{7}} \]
Now we set this slope equal to the slope of the other line, because the lines are parallel and must have the same slope.
\[ \frac{{8 - k}}{{7}} = \frac{{9}}{{4}} \]
This equation helps us find the unknown \( k \) value. By solving this equation, we can determine the y-coordinate of the point \( (-3, k) \) that will make the lines parallel.
Solving Equations
To find the value of \( k \) that makes the slopes equal (and thus the lines parallel), we need to solve the equation:
\[ \frac{{8 - k}}{{7}} = \frac{{9}}{{4}} \]
We can use cross-multiplication to solve this equation. Cross-multiplication involves multiplying the numerator of one fraction by the denominator of the other fraction and vice versa. This clears the fractions and simplifies the equation:
\[ 4(8 - k) = 9(7) \]
\[ 32 - 4k = 63 \]
\[ -4k = 63 - 32 \]
\[ -4k = 95 \]
Finally, divide both sides by -4 to solve for \( k \):
\[ k = \frac{{95}}{{4}} \]
This solution shows that \( k = \frac{{95}}{{4}} \) satisfies the parallel condition for the given lines. Understanding how to solve equations like this is essential for tackling a wide range of mathematical problems.
One App. One Place for Learning.
All the tools & learning materials you need for study success - in one app.
Get started for free
![Problem 77 Find \(k\) so that the line cont... [FREE SOLUTION] (3)](https://i0.wp.com/website-cdn.studysmarter.de/sites/14/2023/12/Rectangle-22721-2.png "Problem 77 Find \(k\) so that the line cont... [FREE SOLUTION] (3)")
Most popular questions from this chapter
Recommended explanations on Math Textbooks
Pure Maths
Read ExplanationDecision Maths
Read ExplanationStatistics
Read ExplanationDiscrete Mathematics
Read ExplanationCalculus
Read ExplanationMechanics Maths
Read ExplanationWhat do you think about this solution?
We value your feedback to improve our textbook solutions.
Study anywhere. Anytime. Across all devices.
Sign-up for free
This website uses cookies to improve your experience. We'll assume you're ok with this, but you can opt-out if you wish. Accept
Privacy & Cookies Policy
Privacy Overview
This website uses cookies to improve your experience while you navigate through the website. Out of these, the cookies that are categorized as necessary are stored on your browser as they are essential for the working of basic functionalities of the website. We also use third-party cookies that help us analyze and understand how you use this website. These cookies will be stored in your browser only with your consent. You also have the option to opt-out of these cookies. But opting out of some of these cookies may affect your browsing experience.
Always Enabled
Necessary cookies are absolutely essential for the website to function properly. This category only includes cookies that ensures basic functionalities and security features of the website. These cookies do not store any personal information.
Any cookies that may not be particularly necessary for the website to function and is used specifically to collect user personal data via analytics, ads, other embedded contents are termed as non-necessary cookies. It is mandatory to procure user consent prior to running these cookies on your website.
![Problem 77 Find \(k\) so that the line cont... [FREE SOLUTION] (2024)](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAQAAAC1HAwCAAAAC0lEQVR42mP8/h8AAvMB+NzmkbcAAAAASUVORK5CYII=)