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Chapter 3: Problem 39

A line passes through the given points. (a) Find the slope of the line. (b) Write the equation of the line in slope-intercept form. $$ \left(4, \frac{3}{8}\right) ;\left(6, \frac{9}{10}\right) $$

Short Answer

Expert verified
Slope: \( \frac{21}{80} \). Equation: \( y = \frac{21}{80}x - \frac{27}{40} \).

Step by step solution

01

Identify the given points

We are given two points: \( (4, \frac{3}{8}) \) and \( (6, \frac{9}{10}) \). Label them as \( (x_1, y_1) \) and \( (x_2, y_2) \), respectively. Thus, \( x_1 = 4 \), \( y_1 = \frac{3}{8} \), \( x_2 = 6 \), and \( y_2 = \frac{9}{10} \).
02

Find the slope

The formula for the slope \( m \) of a line passing through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Plug in the values: \[ m = \frac{\frac{9}{10} - \frac{3}{8}}{6 - 4} \]Convert the fractions to a common denominator to subtract the numerators: \[ \frac{9}{10} = \frac{36}{40} \text{ and } \frac{3}{8} = \frac{15}{40}, \text{ so } m = \frac{\frac{36}{40} - \frac{15}{40}}{2} = \frac{\frac{21}{40}}{2} = \frac{21}{80} \]
03

Write the equation of the line

We use the slope-intercept form of the equation of a line, which is:\[ y = mx + b \]We already have the slope \( m = \frac{21}{80} \). Now use one of the given points to solve for the y-intercept \( b \). Let's use the point \( (4, \frac{3}{8}) \). Plug in the values:\[ \frac{3}{8} = \frac{21}{80}(4) + b \]Simplify and solve for \( b \):\[ \frac{3}{8} = \frac{21 \times 4}{80} + b \rightarrow \frac{3}{8} = \frac{84}{80} + b \rightarrow \frac{3}{8} = \frac{21}{20} + b \rightarrow b = \frac{3}{8} - \frac{21}{20} \]Find a common denominator to subtract the fractions:\[ \frac{3}{8} = \frac{15}{40}, \frac{21}{20} = \frac{42}{40} \rightarrow b = \frac{15}{40} - \frac{42}{40} \rightarrow b = -\frac{27}{40} \]So, the equation of the line is:\[ y = \frac{21}{80}x - \frac{27}{40} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

slope of a line
Understanding the slope of a line is crucial for analyzing how two points are connected on a graph. The slope (often denoted as \( m \)) shows the rate at which the line ascends or descends. In simpler terms, it tells us how much the y-coordinate of a point on the line changes for a one-unit change in the x-coordinate. We can determine the slope using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). This involves finding the vertical change (rise) and dividing it by the horizontal change (run) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \). For instance, if the coordinates given are (4, 3/8) and (6, 9/10), then the differences are \( y_2 - y_1 = 9/10 - 3/8 \) and \( x_2 - x_1 = 6 - 4 \), which we use in our slope formula to find \( m \).
equation of a line
Once we have the slope, we can form the equation of a line in the slope-intercept form, which is expressed as \( y = mx + b \). Here, \( m \) is the slope and \( b \) is the y-intercept, which is the point where the line crosses the y-axis. To find the y-intercept, we use one of the given points on the line, such as (4, 3/8), and plug the values into our equation along with the slope. So, we would solve \( 3/8 = (21/80)(4) + b \) to uncover the value of \( b \). In our example, this results in \( b = -27/40 \). Thus, the equation of the line is \( y = 21/80 x - 27/40 \). This format makes it easy to graph the line and understand its behavior.
fraction subtraction
When dealing with fractions, especially in equations and slopes, being proficient in fraction subtraction is necessary. To subtract fractions, they must have a common denominator. For example, to subtract \( 3/8 \) from \( 9/10 \), we first convert them to have the same denominator. The lowest common denominator for 8 and 10 is 40. Thus, \( 9/10 = 36/40 \) and \( 3/8 = 15/40 \). Subtracting the fractions gives us \( 36/40 - 15/40 = 21/40 \). Such operations are frequently encountered when finding the difference between y-coordinates in the slope formula. Mastering these steps ensures you can handle any fractions you encounter in algebra.

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Most popular questions from this chapter

Use the slope formula to find the slope of the line that passes through the points. \(\left(\frac{1}{3}, \frac{4}{5}\right) ;\left(\frac{5}{3}, \frac{2}{5}\right)\)

Some learning preferences describe how you prefer to receive, think about, and learn new information. These preferences include visual learning, auditory learning, and kinesthetic learning. Many students use more than one of these categories as they learn mathematics. \- Visual learners prefer to see information. Although you definitely listen to your instructor, you also like to see the example on a white board or screen. You may be able to recall a process by visualizing it in your mind; you may learn better by organizing information in charts, tables, diagrams, or pictures. You may prefer the use of colored markers instead of just black. \- Auditory learners prefer to hear information. Although you definitely watch what your instructor is doing, you also like your instructor to explain things aloud as he or she works. You may find it difficult to take notes because you cannot concentrate enough on what is being said while you write. You may learn better if you have the chance to explain things to others. \- Kinesthetic learners prefer to do. You may find it difficult to sit still and just watch and listen; you want to be trying it out. You may find that you must take notes in order to learn. If you only watch and listen, you may understand the concept but not remember it after you leave the classroom. You often learn better if you can show others how to do things. Have you noticed anything that your instructor does while teaching that you find helps you remember what has been taught?

Use the slope formula to find the slope of the line that passes through the points. \((-5,4) ;(-9,-3)\)

Use the slope formula to find the slope of the line that passes through the points. \((-6,-3) ;(-9,-2)\)

Use the slope formula to find the slope of the line that passes through the points. \(\left(0, \frac{4}{5}\right) ;\left(\frac{1}{5}, 0\right)\)
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