Writing Linear Equations

11 Key excerpts on "Writing Linear Equations"

• 

  eBook - PDF

  Elementary Algebra

  • 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.

  Sign up to read

  Learn more about book
• 

  eBook - ePub

  Let's Review Regents: Algebra I Revised Edition

  • Barron's Educational Series, Gary M. Rubinstein(Authors)
  • 2021(Publication Date)
  • Barrons Educational Services

    (Publisher)

  y when the other variable’s value is known.

  Graphing the Solution Set of a Linear Equation That Is in Slope-Intercept Form

  Math Facts

  An equation like y = 2x + 3 or is said to be in

  slope-intercept form

  . In general, an equation of the form y = mx + b is in slope-intercept form with the m representing the slope and the b representing the y-intercept.

  To graph the solution set of a linear equation that is already in slope–intercept form:

  1. Plot the point (0, b), which is on the y-axis. If the equation is y = 2x + 3, plot the point (0, 3). This is known as the y-intercept of the graph.
  2. If the coefficient of the x term is not already a fraction, turn it into a fraction by putting the coefficient in the numerator of a fraction and a 1 in the denominator. If the coefficient is a negative fraction, make the numerator negative and the denominator positive. For the example, y = 2x + 3, the slope, denoted by m, is 2, which gets changed into .
  3. Starting at the y-intercept you already plotted, move right the number in the denominator of the slope. Then, from where you stopped, move up (down if it is negative) the number in the numerator of the slope. For the y = 2x + 3 example, the slope is so from (0, 3), you move to the right 1 unit and then up 2 units to get to the point (1, 5).
  4. Draw a line through the y-intercept and the new point. Put arrows on both sides of the line to indicate that it continues forever on both sides.

  Example 1

  Make a sketch of the solution set of the graph using the slope-intercept process.

  Solution: Since the constant is 5, the y-intercept is (0, 5). Since the coefficient of the x term is ,

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Introductory Algebra

  Concepts with Applications

  • Charles P. McKeague(Author)
  • 2013(Publication Date)
  • XYZ Textbooks

    (Publisher)

  What type of equation has a line that passes through the origin? EXERCISE SET 3.2 247 SCAN TO ACCESS Vocabulary Review Choose the correct words to fill in the blanks below. solution graphing graph first vertical linear ordered pairs straight line second horizontal 1. The number of an ordered pair is called the x-coordinate. 2. The number of an ordered pair is called the y-coordinate. 3. An ordered pair is a to an equation if it satisfies the equation. 4. The process of obtaining a visual picture of all solutions to an equation is called . 5. A equation in two variables is any equation that can be put into the form ax + bx = c. 6. The first step to graph a linear equation in two variables is to find any three that satisfy the equation. 7. The second step to graph a linear equation in two variables is to the three ordered pairs that satisfy the equation. 8. The third step to graph a linear equation in two variables is to draw a through the three ordered pairs that satisfy the equation. 9. Any equation of the form y = b has a line for it’s graph, whereas any equation of the form x = a has a line for it’s graph. Problems A For each equation, complete the given ordered pairs. 1. 2x + y = 6 (0, ), ( , 0), ( , −6) 2. 3x − y = 5 (0, ), (1, ), ( , 5) 3. 3x + 4y = 12 (0, ), ( , 0), (−4, ) 4. 5x − 5y = 20 (0, ), ( , −2), (1, ) 5. y = 4x − 3 (1, ), ( , 0), (5, ) 6. y = 3x − 5 ( , 13), (0, ), (−2, ) 7. y = 7x − 1 (2, ), ( , 6), (0, ) 8. y = 8x + 2 (3, ), ( , 0), ( , −6) 9. x = −5 ( , 4), ( , −3), ( , 0) 10. y = 2 (5, ), (−8, ),  1 __ 2 ,  248 Chapter 3 Graphing Linear Equations and Inequalities in Two Variables For each of the following equations, complete the given table. 11. y = 3x 12. y = −2x 13. y = 4x 14. y = −5x 15. x + y = 5 16. x − y = 8 17. 2x − y = 4 18. 3x − y = 9 19. y = 6x − 1 20. y = 5x + 7 21. y = −2x + 3 22. y = −3x + 1 B For the following equations, tell which of the given ordered pairs are solutions. 23. 2x − 5y = 10 (2, 3), (0, −2),  5 __ 2 , 1  24.

  Sign up to read

  Learn more about book
• 

  eBook - ePub

  Linear Algebra

  An Inquiry-Based Approach

  • Jeff Suzuki(Author)
  • 2021(Publication Date)
  • CRC Press

    (Publisher)

  2

  Systems of Linear Equations

  In this chapter, we’ll consider the problem of solving systems of linear equations. This is a critical part of linear algebra, and we offer the following general strategy:

  Strategy. Every problem in linear algebra begins with a system of linear equations.

  2.1 Standard Form

  When trying to work with a collection of objects, it’s easiest if they’re all in some standardized format. You’ve seen such standardized formats before: it’s convenient to rewrite a given quadratic equation in the form

  a

  x 2

  + b x + c = 0

  ; it’s convenient to write the equation of a line in the form

  y = m x + b

  ; and so on. In this activity, we’ll develop a standard form for a system of linear equations.

  Activity 2.1: Standard Form We’ll say that an equation has been reduced to simplest form if it has as few terms as possible.

  A2.1.1 The equation

  3 x − 5 y = 8

  has three terms: 3x, 5y, and 8. If possible, rewrite the equation so it has fewer terms; if not possible, explain why not.

  A2.1.2 The equation

  2 x + 7 z = 1 + 2 y + 5 x

  has five terms. If possible, rewrite the equation so it has fewer terms; if not possible, explain why not.

  A2.1.3 The equation

  5 x − 2 y + z + 12 = 0

  has five terms. If possible, rewrite the equation so it has fewer terms; if not possible, explain why not.

  Activity 2.1 motivates the following definition:

  Definition 2.1 (Standard Form). A linear equation is in standard form when

  • All of the variables are on one side of the =,
  • The constant is on the other side of the =.

  As Activity 2.1 shows, equations in standard form will have fewer terms than equations not in standard form.

  In Activity A1.3.2, we represented linear equations as vectors. If we have a system of equations, we can represent the entire system as a collection of vectors and, as long as all

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Beginning and Intermediate Algebra

  A Guided Approach

  • Rosemary Karr, Marilyn Massey, R. Gustafson, , Rosemary Karr, Marilyn Massey, R. Gustafson(Authors)
  • 2014(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  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. 190 CHAPTER 3 Graphing; Writing Equations of Lines; Functions; Linear Inequalities in Two Variables m 5 y 2 2 y 1 x 2 2 x 1 This is the slope formula. 5 2 6 2 4 8 2 1 2 5 2 Substitute 2 6 for y 2 , 4 for y 1 , 8 for x 2 , and 2 5 for x 1 . 5 2 10 13 Since the line passes through both points, we can choose either one and substitute its coordinates into the point-slope equation. If we choose 1 2 5, 4 2 , we substitute 2 5 for x 1 , 4 for y 1 , and 2 10 13 for m . y 2 y 1 5 m 1 x 2 x 1 2 This is point-slope form. y 2 4 5 2 10 13 3 x 2 1 2 5 24 Substitute. y 2 4 5 2 10 13 1 x 1 5 2 2 1 2 5 2 5 5 To solve for y , we proceed as follows: y 2 4 5 2 10 13 x 2 50 13 Use the distributive property to remove parentheses. y 5 2 10 13 x 1 2 13 Add 4 to both sides and simplify. The equation is y 5 2 10 13 x 1 2 13 . Find the point-slope equation of the line that passes through 1 2 2, 5 2 and 1 5, 2 2 2 . Then solve it for y . y 5 2 x 1 3 Graph a line given the point-slope form of an equation. Graphing a linear equation written in point-slope form does not require solving for one of the variables. For example, to graph y 2 2 5 3 4 1 x 2 1 2 we compare the equation to point-slope form y 2 y 1 5 m 1 x 2 x 1 2 and note that the slope of the line is m 5 3 4 and that it passes through the point 1 x 1 , y 1 2 5 1 1, 2 2 . Because the slope is 3 4 , we can start at the point 1 1, 2 2 and locate another point on the line by counting 4 units to the right and 3 units up as shown in Figure 3-35.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Algebra

  A Combined Course 2E

  • Charles P. McKeague(Author)
  • 2018(Publication Date)
  • XYZ Textbooks

    (Publisher)

  What type of equation has a line that passes through the origin? 266 Chapter 3 Graphing Linear Equations in Two Variables E X E R C I S E S E T 3.2 VOCABULARY REVIEW Choose the correct words to fill in the blanks below. solution graphing graph first vertical linear ordered pairs straight line second horizontal 1. The number of an ordered pair is called the x-coordinate. 2. The number of an ordered pair is called the y-coordinate. 3. An ordered pair is a to an equation if it satisfies the equation. 4. The process of obtaining a visual picture of all solutions to an equation is called . 5. A equation in two variables is any equation that can be put into the form ax + bx = c. 6. The first step to graph a linear equation in two variables is to find any three that satisfy the equation. 7. The second step to graph a linear equation in two variables is to the three ordered pairs that satisfy the equation. 8. The third step to graph a linear equation in two variables is to draw a through the three ordered pairs that satisfy the equation. 9. Any equation of the form y = b has a line for it’s graph, whereas any equation of the form x = a has a line for it’s graph. A For each equation, complete the given ordered pairs. 1. 2x + y = 6 (0, ), ( , 0), ( , −6) 2. 3x − y = 5 (0, ), (1, ), ( , 5) 3. 3x + 4y = 12 (0, ), ( , 0), (−4, ) 4. 5x − 5y = 20 (0, ), ( , −2), (1, ) 5. y = 4x − 3 (1, ), ( , 0), (5, ) 6. y = 3x − 5 ( , 13), (0, ), (−2, ) 7. y = 7x − 1 (2, ), ( , 6), (0, ) 8. y = 8x + 2 (3, ), ( , 0), ( , −6) 9. x = −5 ( , 4), ( , −3), ( , 0) 10. y = 2 (5, ), (−8, ),  1 __ 2 ,  For each of the following equations, complete the given table. 11. y = 3x 12. y = −2x 13. y = 4x 14. y = −5x x y 1 3 −3 12 18 x y −4 0 10 12 x y 0 −2 −3 12 x y 3 0 −2 −20 3.2 Exercise Set 267 15. x + y = 5 16. x − y = 8 17. 2x − y = 4 18. 3x − y = 9 19. y = 6x − 1 20. y = 5x + 7 21. y = −2x + 3 22. y = −3x + 1 B For the following equations, tell which of the given ordered pairs are solutions.

  Sign up to read

  Learn more about book
• 

  eBook - ePub

  Precalculus

  A Self-Teaching Guide

  • Steve Slavin, Ginny Crisonino(Authors)
  • 2001(Publication Date)
  • Trade Paper Press

    (Publisher)

  m = −2. We can use the first formula:

  2. First let’s start by finding the slope

  We don’t know the y-intercept, so we have to use the second formula:

  3. We don’t know the y-intercept, so we’ll use the second formula:

  4. The points we’re given are (6,0) and (0,−3). We need to find the slope.

  5. We know this is a horizontal line because both points have the same y value. We also know that the equation of any horizontal line is y = a constant, y = 1 is the equation of this horizontal line.

  6. We know this is a vertical line because both points have the same x value. We also know that the equation of any vertical line is x = a constant, x = 5 is the equation of this vertical line.

  4 Graphs of Linear Functions

  A linear equation is a polynomial equation whose highest exponent is 1. It’s also called a first-degree equation. The graph of a linear equation is always a straight line. We need only two points to graph a straight line. The points we want to see on your graphs are the intercepts. You’ll find graphing linear equations to be very simple, especially since we reviewed how to find intercepts in section 1 of this chapter.

  Example 10:

  Graph 2x + 4y = 8.

  Solution:

  We’ll start by finding the x and the y intercepts; then we’ll plot the points and connect them with a straight line. Be sure to extend the line beyond those points and put arrows at the ends of the lines to show that the graph doesn’t end at these points. If you don’t put arrows at the end, it’s a line segment, which is only part of the graph. Write the equation of the line on your graph and label the intercepts.

  Example 11:

  Graph −2x + 4y = 8.

  Solution:

  Notice that this equation is almost identical to the equation in example 10. The only difference is the negative in front of the 2x. Let’s see what kind of an effect this has on our graph.

  The negative in front of the 2x reverses the line’s direction. When we write the equation −2x + 4y = 8 in y = mx + b form, it’s . Now it’s rising instead of falling. The slope of the line in example 9 is positive and its y-intercept is 2. When we write the equation 2x + 4y = 8 in y = mx + b form it’s . The slope of the line in example 8 is negative and its y

  Sign up to read

  Learn more about book
• 

  eBook - ePub

  Making Sense of Mathematics for Teaching Grades 6-8

  (Unifying Topics for an Understanding of Functions, Statistics, and Probability)

  • Edward C. Nolan, Juli K. Dixon(Authors)
  • 2016(Publication Date)
  • Solution Tree Press

    (Publisher)

  Much of the focus of the content of expressions, equations, and inequalities in the middle grades provides the groundwork for understanding algebra. As students make sense of real-world situations with unknown quantities that do and do not vary, they engage in algebraic reasoning. Students learn to write and solve expressions, equations, and inequalities, using context to help make sense of the language and structure of mathematics. Students in grades 6–8 must be able to write expressions and equations from context, solve equations and inequalities, investigate the concept of slope, and solve systems of equations.

  Writing Expressions and Equations From Context

  In language, an expression communicates an idea. The same can be said for algebraic expressions—they communicate what contexts represent. Interpreting context lays a foundation for making sense of algebra. Consider the task provided in figure 3.7 .

  You are cleaning out your desk at school. You find that you have 6 times as many pencils as highlighters. Write an equation for the number of pencils (p ) in terms of the number of highlighters (h ).

  Figure 3.7: The pencils and highlighters task.

  How did you set up your equation? When posed with this task, many people write the equation 6p = h . How can you test this equation to see if it matches the context? When you start to substitute numbers for the variables in the equation 6p = h , what do you notice? If you let p equal 12, then h is 72. Does this make sense in the context of the problem? Returning to the context, you realize that there should be more pencils than highlighters, in fact, six times more. Instead of thinking about the relationship between the number of pencils and the number of highlighters, the equation 6p = h was written to follow the order of the items presented in the task; whereas, the equation 6h = p correctly represents the mathematical relationship. This is an important example of the need to combine the language of mathematics—and possibly the use of the key words

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Introductory Algebra

  Concepts and Graphs 2E

  • Charles P. McKeague(Author)
  • 2020(Publication Date)
  • XYZ Textbooks

    (Publisher)

  B Find an equation of a line using the point-slope form. 3.6 VIDEOS 218 CHAPTER 3 Linear Equations and Inequalities in Two Variables y − 1 _____ x − 0 = 3 __ 2 Slope = vertical change __________ horizontal change y − 1 _____ x = 3 __ 2 x − 0 = x y − 1 = 3 __ 2 x Multiply each side by x y = 3 __ 2 x + 1 Add 1 to each side What is interesting and useful about the equation we have just found is that the number in front of x is the slope of the line and the constant term is the y-intercept. It is no coincidence that it turned out this way. Whenever an equation has the form y = mx + b, the graph is always a straight line with slope m and y-intercept b. To see that this is true in general, suppose we want the equation of a line with slope m and y-intercept b. Because the y-intercept is b, then the point (0, b ) is on the line. If (x, y) is any other point on the line, then we apply our slope formula to get y − b _____ x − 0 = m Slope = vertical change __________ horizontal change y − b _____ x = m x − 0 = x y − b = mx Multiply each side by x y = mx + b Add b to each side x y (x, y) (0, 1) y - 1 x - 0 x y (x, y) (0, b) y - b x - 0 3.6 Finding the Equation of a Line 219 Here is a summary of what we have just found. The equation of the line with slope m and y-intercept b is always given by y = mx + b PROPERTY: SLOPE-INTERCEPT FORM OF THE EQUATION OF A LINE Find the equation of the line with slope − 4 _ 3 and y-inter- cept 5. Then, graph the line. SOLUTION Substituting m = − 4 _ 3 and b = 5 into the equation y = mx + b, we have y = − 4 __ 3 x + 5 Finding the equation from the slope and y-intercept is just that easy. If the slope is m and the y-intercept is b, then the equation is always y = mx + b. Because the y-intercept is 5, the graph goes through the point (0, 5). To find a second point on the graph, we start at (0, 5) and move 4 units down (that’s a rise of −4) and 3 units to the right (a run of 3).

  Sign up to read

  Learn more about book
• 

  eBook - ePub

  Differentiating Instruction in Algebra 1

  Ready-to-Use Activities for All Students (Grades 7-10)

  • Kelli Jurek(Author)
  • 2021(Publication Date)
  • Routledge

    (Publisher)

  y-intercept. Their finished puzzle is the message "Math Does Rock Says Me 2 U" (with the help of some text messaging shortcuts).
• ➤ Students may notice that there are puzzle pieces that are blank on one side. They may figure out that these pieces belong on the outer edge of the puzzle.
Discussion Questions (To be asked after the lesson.)
• ➤ What message is revealed when the puzzle is completed?
• ➤ How do you determine whether an ordered pair is a solution to an equation? How many solutions are there to a linear equation?
• ➤ Can all equations be rewritten in the form of y =? Why would we want to write equations in this form?
• ➤ How do you determine the slope and the y-intercept from an equation?

Lesson 1: IntRoduction

Student Coordinate Grid Whole-Class Activity Lesson Plan

Purpose: Students will model a system of linear equations by using the classroom as a coordinate grid.

Prerequisite Knowledge	Materials Needed
Students should know how to:• solve equations,• check solutions to equations,• graph equations on calculators, and• find coordinates on a coordinate grid.	• One copy of the student worksheet for each student• Note cards with coordinate pairs (0, 0) through (5, 4)• Overhead projector with 6x5 grid matching the setup of the desks

Discussion Questions (To be asked before the lesson.)

• ➤ Can two different linear equations have one solution in common?
• ➤ What does this solution mean?

Lesson

• ➤ Write the following equations on the board, and label them for future reference.
  Equation A x + y= 5
  Equation B x- y= -1
• ➤ Arrange the classroom desks into a coordinate grid. Each student will represent one of the ordered pairs. Below is an example of a class with 30 students. This activity could also be done outside with a grid drawn with sidewalk chalk.