Mathematics
What is Point Slope Form
Point-slope form is a method used to find the equation of a straight line when given the slope and one point on the line. It is written as y - y1 = m(x - x1), where m is the slope and (x1, y1) is the given point. This form is useful in graphing and solving problems involving linear equations.
Written by Perlego with AI-assistance
Related key terms
1 of 5
11 Key excerpts on "What is Point Slope Form"
eBook - PDFPractical Algebra
A Self-Teaching Guide
- Bobson Wong, Larisa Bukalov, Steve Slavin(Authors)
- 2022(Publication Date)
- Jossey-Bass(Publisher)
In general, the point-slope form of the equation of a line is y − y 1 = m ( x − x 1 ) , where m is the slope and ( x 1 , y 1 ) is a point on the line. The following example shows why the point-slope form is useful. Example 7.12 Write an equation of the line that has a slope of − 𝟐 𝟓 and passes through (6, − 5). Solution: We can use either point-slope or slope-intercept form of the equation. METHOD 1: Use point-slope form . Here, m = − 2 5 , x 1 = 6, and y 1 = − 5. When we substitute into the formula for point-slope form, we get y − (− 5 ) = − 2 5 ( x − 6 ) , or y + 5 = − 2 5 ( x − 6 ) . 188 PRACTICAL ALGEBRA METHOD 2: Use slope-intercept form . The equation in slope-intercept form is y = mx + b . Here, m = − 2 5 . Since x and y represent the set of all ordered pairs ( x , y ) that satisfy the equation, we can substitute the ordered pair (6, − 5). We substitute x = 6 and y = − 5 into the slope-intercept form to solve for b : y = mx + b − 5 = − 2 5 ( 6 ) + b − 5 = − 12 5 + b Multiply. + 12 5 + 12 5 Add 12 5 to both sides. − 13 5 = b Answer, which we write as b = − 13 5 . The equation of the line in slope-intercept form is y = − 2 5 x + ( − 13 5 ) , or y = − 2 5 x − 13 5 . Although these two equations look different, they are actually equiva-lent. We can rearrange the point-slope form into slope-intercept form by distributing and rearranging terms: y + 5 = − 2 5 ( x − 6 ) y + 5 = − 2 5 x − 2 5 (− 6 ) Distributive property. y + 5 = − 2 5 x + 12 5 Multiply. − 5 − 5 Subtract 5 from both sides. y = − 2 5 x − 13 5 Final answer. As you can see, the point-slope form of a line can simplify our work tremendously! To write the equation of a line in point-slope or slope-intercept form, we need the slope of the line. If we’re not given the slope—for example, if we’re only given the coordinates of two points on the line—then we use the coordinates to find the slope.
eBook - PDF- Alan Tussy, R. Gustafson(Authors)
- 2012(Publication Date)
- Cengage Learning EMEA(Publisher)
Of all of the ways in which a linear equation can be written, one form, called slope – intercept form, is probably the most useful. When an equation is written in this form, two important features of its graph are evident. Use Slope–Intercept Form to Identify the Slope and y -Intercept of a Line. To explore the relationship between a linear equation and its graph, let’s consider . To graph this equation, three values of x were selected ( 1, 0, and 1), the corresponding values of y were found, and the results were entered in the table. Then the ordered pairs were plotted and a straight line was drawn through them, as shown below. y 2 x 1 y 2 x 1 0 1 1 3 (1, 3) (0, 1) ( 1, 1) 1 1 ( x , y ) y x To find the slope of the line, we pick two points on the line, and , and draw a slope triangle and count grid squares: Slope rise run 2 1 2 (0, 1) ( 1, 1) x y y = 2 x + 1 2 1 (–1, –1) (0, 1) (1, 3) 1 –1 2 3 4 –2 –3 –4 –2 –3 –4 2 3 4 Copyright 201 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. From the equation and the graph, we can make two observations: ■ The graph crosses the -axis at 1. This is the same as the constant term in . ■ The slope of the line is 2. This is the same as the coefficient of in . This illustrates that the slope and -intercept of the graph of can be determined from the equation. The slope of the line is 2. The -intercept is . These observations suggest the following form of an equation of a line.
eBook - PDFCollege Algebra
Building Skills and Modeling Situations
- Charles P. McKeague, Katherine Yoshiwara, Denny Burzynski(Authors)
- 2013(Publication Date)
- XYZ Textbooks(Publisher)
282 Chapter 4 Slope, Rates of Change, and Linear Functions This form of the equation of a line is used to find the equation of a line, either given one point on the line and the slope, or given two points on the line. Find the equation of the line with slope −2 that contains the point (−4, 3). Write the answer in slope-intercept form. SOLUTION Using (x 1 , y 1 ) = (−4, 3) and m = −2 in y − y 1 = m(x − x 1 ) Point-slope form gives us y − 3 = −2(x + 4) Note: x − (−4) = x + 4 y − 3 = −2x − 8 Multiply out right side y = −2x − 5 Add 3 to each side Figure 4 is the graph of the line that contains (−4, 3) and has a slope of −2. Notice that the y-intercept on the graph matches that of the equation we found. Find the equation of the line that passes through the points (−3, 3) and (3, −1). SOLUTION We begin by finding the slope of the line: m = 3 − (−1) ________ −3 − 3 = 4 ___ −6 = − 2 __ 3 PROPERTY Point-Slope Form of the Equation of a Line The equation of the line through (x 1 , y 1 ) with slope m is given by y − y 1 = m(x − x 1 ) EXAMPLE 4 FIGURE 4 x y 2 1 –1 –3 –4 –5 3 4 5 1 –3 –5 2 3 4 5 –2 –1 1 –1 –3 –4 –5 3 4 5 1 –3 –5 2 3 4 5 –2 –1 1 1 Slope = − 2 __ 1 = −2 (–4, 3) (0, –5) –2 y-intercept = −5 EXAMPLE 5 4.2 Linear Functions and Equations of Lines 283 Using (x 1 , y 1 ) = (3, −1) and m = − 2 _ 3 in y − y 1 = m(x − x 1 ) yields y + 1 = − 2 __ 3 (x − 3) y + 1 = − 2 __ 3 x + 2 Multiply out right side y = − 2 __ 3 x + 1 Add −1 to each side Figure 5 shows the graph of the line that passes through the points (−3, 3) and (3, −1). As you can see, the slope and y-intercept are − 2 _ 3 and 1, respectively. The last form of the equation of a line that we will consider in this section is called the standard form. It is used mainly to write equations in a form that is free of fractions and is easy to compare with other equations.
eBook - PDF- Mark D. Turner, Charles P. McKeague(Authors)
- 2016(Publication Date)
- XYZ Textbooks(Publisher)
188 CHAPTER 3 Linear Equations in Two Variables and Functions This last equation is known as the point-slope form of the equation of a line. This form of the equation of a line is used to find the equation of a line, either given one point on the line and the slope, or given two points on the line. Find the equation of the line with slope − 2 that contains the point ( − 4, 3). Write the answer in slope-intercept form. SOLUTION Using ( x 1 , y 1 ) = ( − 4, 3) and m = − 2 in y − y 1 = m ( x − x 1 ) Point-slope form gives us y − 3 = − 2( x + 4) Note: x − ( − 4) = x + 4 y − 3 = − 2 x − 8 Multiply out right side y = − 2 x − 5 Add 3 to each side Figure 3 is the graph of the line that contains ( − 4, 3) and has a slope of − 2. Notice that the y -intercept on the graph matches that of the equation we found. Find the equation of the line that passes through the points ( − 3, 3) and (3, − 1). SOLUTION We begin by finding the slope of the line: m = 3 − ( − 1) ________ − 3 − 3 = 4 ___ − 6 = − 2 __ 3 Using ( x 1 , y 1 ) = (3, − 1) and m = − 2 _ 3 in y − y 1 = m ( x − x 1 ) yields y + 1 = − 2 __ 3 ( x − 3) y + 1 = − 2 __ 3 x + 2 Multiply out right side y = − 2 __ 3 x + 1 Add − 1 to each side PROPERTY Point-Slope Form of the Equation of a Line The equation of the line through ( x 1 , y 1 ) with slope m is given by y − y 1 = m ( x − x 1 ) EXAMPLE 5 FIGURE 3 x y 2 1 –1 –3 –4 –5 3 4 5 1 –3 –5 2 3 4 5 –2 –1 1 –1 –3 –4 –5 3 4 5 1 –3 –5 2 3 4 5 –2 –1 Slope H11005 H11002 2 1 H11005 H11002 2 y -intercept H11005 H11002 5 (0, H11002 5) ( H11002 4, 3) 1 H11002 2 Slope H11005 H11002 2 1 H11005 H11002 2 y -intercept H11005 H11002 5 (0, H11002 5) ( H11002 4, 3) 1 H11002 2 EXAMPLE 6 Note We could have used the point ( − 3, 3) instead of (3, − 1) and obtained the same equation. That is, using ( x 1 , y 1 ) = ( − 3, 3) and m = − 2 _ 3 in y − y 1 = m ( x − x 1 ) gives us y − 3 = − 2 _ 3 ( x + 3) y − 3 = − 2 _ 3 x − 2 y = − 2 _ 3 x + 1 which is the same result we obtained using (3, − 1).
eBook - PDF- Ron Larson, Bruce Edwards(Authors)
- 2018(Publication Date)
- Cengage Learning EMEA(Publisher)
16 Chapter 1 Preparation for Calculus 1.2 Exercises See CalcChat.com for tutorial help and worked-out solutions to odd-numbered exercises. CONCEPT CHECK 1. Slope-Intercept Form In the form y = mx + b, what does m represent? What does b represent? 2. Perpendicular Lines Is it possible for two lines with positive slopes to be perpendicular? Why or why not? Estimating Slope In Exercises 3–6, estimate the slope of the line from its graph. To print an enlarged copy of the graph, go to MathGraphs.com. 3. x 1 2 3 4 5 6 7 7 6 5 4 3 2 1 y 4. x 1 2 3 4 5 6 7 7 6 5 3 2 1 y 5. x 1 2 3 4 5 6 6 5 4 3 2 1 y 6. x 1 2 3 5 6 7 24 28 20 16 12 8 4 y Finding the Slope of a Line In Exercises 7–12, plot the pair of points and find the slope of the line passing through them. 7. (3, -4), (5, 2) 8. (0, 0), (-2, 3) 9. (4, 6), (4, 1) 10. (3, -5), (5, -5) 11. (- 1 2 , 2 3 ), (- 3 4 , 1 6 ) 12. ( 7 8 , 3 4 ), ( 5 4 , - 1 4 ) Sketching Lines In Exercises 13 and 14, sketch the lines through the point with the indicated slopes. Make the sketches on the same set of coordinate axes. Point Slopes 13. (3, 4) (a) 1 (b) -2 (c) - 3 2 (d) Undefined 14. (-2, 5) (a) 3 (b) -3 (c) 1 3 (d) 0 Finding Points on a Line In Exercises 15–18, use the point on the line and the slope of the line to find three additional points that the line passes through. (There is more than one correct answer.) Point Slope Point Slope 15. (6, 2) m = 0 16. (-4, 3) m is undefined. 17. (1, 7) m = -3 18. (-2, -2) m = 2 Finding an Equation of a Line In Exercises 19–24, find an equation of the line that passes through the point and has the indicated slope. Then sketch the line. Point Slope 19. (0, 3) m = 3 4 20. (-5, -2) m = 6 5 21. (1, 2) m is undefined. 22. (0, 4) m = 0 23. (3, -2) m = 3 24. (-2, 4) m = - 3 5 25. Road Grade You are driving on a road that has a 6% uphill grade. This means that the slope of the road is 6 100 . Approximate the amount of vertical change in your position when you drive 200 meters.
No longer available |Learn more- Alfred Basta, Stephan DeLong, Nadine Basta, , Alfred Basta, Stephan DeLong, Nadine Basta(Authors)
- 2013(Publication Date)
- Cengage Learning EMEA(Publisher)
This particular form of the equation of a line is fairly restrictive and serves as a point of agreement where equations in different forms can be compared for equivalence without risk of ambiguity. Now that we’ve introduced a general form, it is natural to wonder what other named forms of linear equations exist. In turn, we will examine various types, including the slope-intercept form, the point-slope form, the two-point form, and the two-intercept form. You might suspect that one form would be the preferred form, but you would be mistaken. Each of the various forms has a particular usefulness, and having knowledge of all of them will make our ex-amination of particular types of problems easier. Remember that it is always good to have a variety of tools at our disposal because some methods might be more effective than others for specific exercises. Were we to acknowledge the existence of only one form of the equation of a line, we would be severely limit-ing the efficiency of our work! Now, let’s return to the general form we defined and pose the following question. Why do equations of the form Ax 1 By 1 C 5 0 represent straight lines? The graph of any equation consists of the set of all points ( x , y ) in the plane whose coordinates satisfy the equation. That is, substitution of the coordi-nate values into the equation creates a true equality. We can produce the graph of the equation by constructing a table of values, plotting the points obtained from that table, and sketching a graph that follows the perceived pattern dem-onstrated by those points. Let’s do that now, for the example 3 x 2 5 y 5 25. Choosing some values for x , such as 0, 5, 10, and 15, we can produce Table 4.1. TABLE 4.1 Table of values x y 0 2 5 5 2 2 10 1 15 4 The values of y are found by inserting our arbitrarily chosen values of x into the equation and solving for y .
eBook - ePub- Barron's Educational Series, Amy Stahl(Authors)
- 2021(Publication Date)
- Barrons Educational Services(Publisher)
y-value will always be 2. Then plot the points, and draw the line through the points. This is a horizontal line.x y −2 2 0 2 3 2 Graph x = −1.Make a table, but this time the x-values always stay the same. The x-value has to be −1. Choose three different y-values to help you make the graph. Plot the points, and draw the line. This is a vertical line.x y −1 −2 −1 0 −1 2 CAUTION—Major Mistake Territory!A horizontal line is always written in the form y = b. A vertical line is always written in the form x = a.Examples:y = 3 → Horizontal linex = −2 → Vertical lineSlope of a LineBRAIN TICKLERSSet # 17Graph each line.1.x = 42.x = −23.y = 54.y = −1(Answers are on pages 117 –118 .)Have you ever tried to run up a steep hill? Or ride a bike all the way up a hill without walking? Or ski down a hill? The concept of “steepness” is one that can easily be understood in the real world.Examples of different “slopes” are easy to see. As a beginner skier, which hill would you prefer to ski down? A beginner skier starts on the bunny hill, which is less steep than a hill considered a black diamond!In the world of mathematics, the word slope describes the “steepness” of a line.There are four basic types of slope.
The slope of a line is described as a ratio. It is the comparison of the vertical change in the line compared to the horizontal change in the line. Here is the formal definition of slope.Positive slopeSlants upward, from left to rightNegative slopeSlants downward, from left to rightZero slopeA horizontal line, left to right Has zero slope (like a floor)No slope/Undefined slopeA vertical line, up and down Has undefined slope (like a wall)For a painless way to remember slope, use this saying:•Rise is the vertical distance between points. Rise involves the y-
eBook - PDF- Lynn Marecek, MaryAnne Anthony-Smith, Andrea Honeycutt Mathis(Authors)
- 2020(Publication Date)
- Openstax(Publisher)
Chapter 4 Graphs 495 Writing Exercises 285. What does the sign of the slope tell you about a line? 286. How does the graph of a line with slope m = 1 2 differ from the graph of a line with slope m = 2 ? 287. Why is the slope of a vertical line “undefined”? Self Check ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section. ⓑ On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this? 496 Chapter 4 Graphs This OpenStax book is available for free at http://cnx.org/content/col31130/1.4 4.5 Use the Slope-Intercept Form of an Equation of a Line Learning Objectives BE PREPARED : : 4.10 Before you get started, take this readiness quiz. Add: x 4 + 1 4 . If you missed this problem, review Example 1.77. BE PREPARED : : 4.11 Find the reciprocal of 3 7 . If you missed this problem, review Example 1.70. BE PREPARED : : 4.12 Solve 2x − 3y = 12 for y . If you missed this problem, review Example 2.63. Recognize the Relation Between the Graph and the Slope–Intercept Form of an Equation of a Line We have graphed linear equations by plotting points, using intercepts, recognizing horizontal and vertical lines, and using the point–slope method. Once we see how an equation in slope–intercept form and its graph are related, we’ll have one more method we can use to graph lines. In Graph Linear Equations in Two Variables, we graphed the line of the equation y = 1 2 x + 3 by plotting points. See Figure 4.24. Let’s find the slope of this line. Figure 4.24 The red lines show us the rise is 1 and the run is 2. Substituting into the slope formula: m = rise run m = 1 2 What is the y-intercept of the line? The y-intercept is where the line crosses the y-axis, so y-intercept is (0, 3) . The equation of this line is: Chapter 4 Graphs 497
eBook - PDFBeginning Algebra
Connecting Concepts through Applications
- Mark Clark, Cynthia Anfinson(Authors)
- 2018(Publication Date)
- Cengage Learning EMEA(Publisher)
Using two points on the line, find the slope m . 2. To find the y -intercept (0, b ), substitute any point on the line in for x and y in y 5 mx 1 b and solve for b . 3. Write the equation of the line y 5 mx 1 b with the specific values for m and b found in steps 1 and 2. ● ● The point-slope form of a line is y 2 y 1 5 m 1 x 2 x 1 2 , where 1 x 1 , y 1 2 is any point on the line and m is the slope of the line. Find the equation of the line that passes through the points (5, 1) and 1 2 5, 2 3 2 . The slope of the line is m 5 2 3 2 1 2 5 2 5 5 2 4 2 10 5 2 5 Substituting this result into y 5 mx 1 b yields y 5 2 5 x 1 b . To find the value of b , substitute either point into the equation for x and y and solve for b . 1 5 2 5 1 5 2 1 b 1 5 2 1 b 2 2 2 2 2 1 5 b Therefore, the equation of the line is y 5 2 5 x 2 1 . Example 10 SOLUTION Substitute in 1 x , y 2 5 1 5, 1 2 . Copyright 2019 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. C H A P T E R 3 L i n e a r E q u a t i o n s w i t h T w o V a r i a b l e s 346 Using point-slope form, find the equation of the line passing through the points 1 1, 2 3 2 and 1 2 2, 2 15 2 . Put the final answer in slope-intercept form, y 5 mx 1 b . The slope of the line is m 5 2 15 2 1 2 3 2 2 2 2 1 5 2 12 2 3 5 4 Substituting the slope and the point 1 x 1 , y 1 2 5 1 1, 2 3 2 into the point-slope form yields y 2 1 2 3 2 5 4 1 x 2 1 2 y 1 3 5 4 x 2 4 2 3 2 3 y 5 4 x 2 7 Example 11 SOLUTION Section 3.6 Modeling Linear Data ● ● Scatterplots can be used to look for patterns in data.
eBook - PDF- Lynn Marecek, MaryAnne Anthony-Smith, Andrea Honeycutt Mathis(Authors)
- 2020(Publication Date)
- Openstax(Publisher)
Division by 0 is undefined. So we say that the slope of the vertical line x = 3 is undefined. The slope of all vertical lines is undefined, because the run is 0. Slope of a Vertical Line The slope of a vertical line, x = a, is undefined. EXAMPLE 11.37 Find the slope of each line: 1024 Chapter 11 Graphs This OpenStax book is available for free at http://cnx.org/content/col30939/1.6 ⓐ x = 8 ⓑ y = −5 Solution ⓐ x = 8 This is a vertical line, so its slope is undefined. ⓑ y = −5 This is a horizontal line, so its slope is 0. TRY IT : : 11.72 Find the slope of the line: x = −4. TRY IT : : 11.73 Find the slope of the line: y = 7. Quick Guide to the Slopes of Lines Use the Slope Formula to find the Slope of a Line between Two Points Sometimes we need to find the slope of a line between two points and we might not have a graph to count out the rise and the run. We could plot the points on grid paper, then count out the rise and the run, but there is a way to find the slope without graphing. Before we get to it, we need to introduce some new algebraic notation. We have seen that an ordered pair (x, y) gives the coordinates of a point. But when we work with slopes, we use two points. How can the same symbol (x, y) be used to represent two different points? Mathematicians use subscripts to distinguish between the points. A subscript is a small number written to the right of, and a little lower than, a variable. • (x 1 , y 1 ) read x sub 1, y sub 1 • (x 2 , y 2 ) read x sub 2, y sub 2 We will use (x 1 , y 1 ) to identify the first point and (x 2 , y 2 ) to identify the second point. If we had more than two points, we could use (x 3 , y 3 ), (x 4 , y 4 ), and so on. To see how the rise and run relate to the coordinates of the two points, let’s take another look at the slope of the line between the points (2, 3) and (7, 6) in Figure 11.26. Figure 11.26 Since we have two points, we will use subscript notation. Chapter 11 Graphs 1025
eBook - PDFIntermediate Algebra
A Guided Approach
- Rosemary Karr, Marilyn Massey, R. Gustafson, , Rosemary Karr, Marilyn Massey, R. Gustafson(Authors)
- 2014(Publication Date)
- Cengage Learning EMEA(Publisher)
2 y 2 7 5 1 Determine whether the lines with the given slopes are parallel, perpendicular, or neither. 21. m 1 5 4 , m 2 5 2 1 4 22. m 1 5 0.5 , m 2 5 1 2 23. m 1 5 0.5 , m 2 5 2 1 2 24. m 1 5 5 , m 2 5 2 0.2 25. Sales growth If the sales of a new business were $65,000 in its first year and $130,000 in its fourth year, find the rate of growth in sales per year. DEFINITIONS AND CONCEPTS EXAMPLES Equations of a line: Point-slope form: y 2 y 1 5 m 1 x 2 x 1 2 Write the point-slope form of a line passing through 1 2 3, 5 2 with slope 1 2 . y 2 y 1 5 m 1 x 2 x 1 2 This is point-slope form. y 2 5 5 1 2 3 x 2 1 2 3 24 Substitute. y 2 5 5 1 2 1 x 1 3 2 SECTION 2.3 Writing Equations of Lines Unless otherwise noted, all content on this page is © Cengage Learning. Copyright 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 158 CHAPTER 2 Graphs, Equations of Lines, and Functions Slope-intercept form: y 5 mx 1 b Write the slope-intercept form of a line with slope 1 2 and y -intercept 1 0, 2 4 2 . y 5 m x 1 b This is slope-intercept form. y 5 1 2 x 2 4 Substitute. y 5 1 2 x 2 4 To find the slope of a linear equation written in gen-eral form, solve for y . Find the slope of the line 3 x 1 5 y 5 7. 5 y 5 2 3 x 1 7 y 5 2 3 5 x 1 7 5 The slope is 2 3 5 . REVIEW EXERCISES Write the equation of the line with the given properties in slope-intercept form. 26. slope of 3; passing through P 1 2 8, 5 2 27. passing through 1 2 2, 4 2 and 1 6, 2 9 2 28. passing through 1 2 3, 2 5 2 ; parallel to the graph of 3 x 2 2 y 5 7 29.
Index pages curate the most relevant extracts from our library of academic textbooks. They’ve been created using an in-house natural language model (NLM), each adding context and meaning to key research topics.
