EQUATIONS OF LINES IN SLOPE INTERCEPT AND STANDARD FORM


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1 . Equations of Lines in SlopeIntercept and Standard Form ( ) 8 In this SlopeIntercept Form Standard Form section Using SlopeIntercept Form for Graphing Writing the Equation for a Line Applications (0, ) FIGURE. stud 0 tip (, ) Keep reviewing. After ou have done our current assignment, go back a section or two and tr a few problems. You will be amazed at how much our knowledge will improve with a regular review.. EQUATIONS OF LINES IN SLOPE INTERCEPT AND STANDARD FORM In Section. ou learned that the graph of all solutions to a linear equation in two variables is a straight line. In this section we start with a line or a description of a line and write an equation for the line. The equation of a line in an form is called a linear equation in two variables. SlopeIntercept Form Consider the line through (0, ) with slope shown in Fig... If we use the points (, ) and (0, ) in the slope formula, we get an equation that is satisfied b ever point on the line: m Slope formula 0 Let (, ) (0, ) and (, ) (, ). Now solve the equation for : Multipl each side b. Add to each side. Because (0, ) is on the ais, it is called the intercept of the line. Note how the slope and the coordinate of the intercept (0, ) appear in. For this reason it is called the slopeintercept form of the equation of the line. SlopeIntercept Form The equation of the line with intercept (0, b) and slope m is m b. E X A M P L E Using slopeintercept form Write the equation of each line in slopeintercept form. a) b) c) (, ) (0, ) (0, 0) (, ) (0, ) (, )
2 8 ( 6) Chapter Linear Equations in Two Variables and Their Graphs calculator closeup Since a graphing calculator screen has a finite number of piels, a graphing calculator plots onl a finite number of ordered pairs that satisf an equation. A graph is supposed to be a picture of all of the ordered pairs that satisf an equation. Because a calculator plots man ordered pairs accuratel and quickl, it can be a great help in drawing the graph of an equation. a) The intercept is (0, ), and the slope is. Use the form m b with b andm. The equation in slopeintercept form is. b) The intercept is (0, 0), and the slope is. So the equation is. c) The intercept is (0, ), and the slope is. So the equation is. The equation of a line ma take man different forms. The easiest wa to find the slope and intercept for a line is to rewrite the equation in slopeintercept form. E X A M P L E Finding slope and intercept Determine the slope and intercept of the line 6. Solve for to get slopeintercept form: 6 6 The slope is, and the intercept is (0, ). helpful hint Standard Form The graph of the equation is a vertical line. Because slope is not defined for vertical lines, this line does not have an equation in slopeintercept form. Onl nonvertical lines have equations in slopeintercept form. However, there is a form that includes all lines. It is called standard form. In geometr we learn that two points determine a line. However, if ou locate two points that are close together and draw a line through them, our line can have a lot of error in it at locations far from the two chosen points. Locating five points on the graph will improve our accurac. If ou draw a straight line with two points, the should be chosen as far as possible from each other. Standard Form Ever line has an equation in the form A B C where A, B, and C are real numbers with A and B not both zero. To write the equation in this form, let A, B 0, and C. We get 0, which is equivalent to. In Eample we converted an equation in standard form to slopeintercept form. An linear equation in standard form with B 0 can be written in slopeintercept
3 . Equations of Lines in SlopeIntercept and Standard Form ( 7) 8 form b solving for. In the net eample we convert an equation in slopeintercept form to standard form. E X A M P L E Converting to standard form Write the equation of the line in standard form using onl integers. To get standard form, first subtract from each side: Multipl each side b to eliminate the fraction and get positive. The answer in Eample is not the onl answer using onl integers. Equations such as and 0 0 are equivalent equations in standard form. We prefer to write because the greatest common factor of,, and is and the coefficient of is positive. Using SlopeIntercept Form for Graphing One wa to graph a linear equation is to find several points that satisf the equation and then draw a straight line through them. We can also graph a linear equation b using the intercept and the slope. Strateg for Graphing a Line Using Slope and Intercept. Write the equation in slopeintercept form if necessar.. Plot the intercept.. Starting from the intercept, use the rise and run to locate a second point.. Draw a line through the two points. E X A M P L E To check Eample, graph ( ) on a graphing calculator as follows: calculator closeup The calculator graph is consistent with the graph in Fig... Graphing a line using intercept and slope Graph the line. First write the equation in slopeintercept form: Subtract from each side. Divide each side b. The slope is, and the intercept is (0, ). A slope of means a rise of and a run of. Start at (0, ) and go up two units and to the right three units to locate a second point on the line. Now draw a line through the two points. See Fig.. for the graph of. = Run = Rise = FIGURE.
4 8 ( 8) Chapter Linear Equations in Two Variables and Their Graphs CAUTION When using the slope to find a second point on the line, be sure to start at the intercept, not at the origin. E X A M P L E Graphing a line using intercept and slope Graph the line. The slope is, and the intercept is (0, ). Because, we use a rise of and a run of. To locate a second point on the line, start at (0, ) and go down three units and to the right one unit. Draw a line through the two points. See Fig... (0, ) = + + FIGURE. Writing the Equation for a Line In Eample we wrote the equation of a line b finding its slope and intercept from a graph. In the net eample we write the equation of a line from a description of the line. E X A M P L E 6 Writing an equation Write the equation in slopeintercept form for the line through (0, ) that is perpendicular to the line. First find the slope of : The slope of this line is. The slope of the line that we are interested in is the opposite of the reciprocal of. So the line has slope andintercept (0, ). Its equation is. calculator closeup If ou use the same minimum and maimum window values for and, then the length of one unit on the ais is larger than on the ais because the screen is longer in the direction. In this case, perpendicular lines will not look perpendicular. The viewing window chosen here for the lines in Eample 6 makes them look perpendicular. An viewing window proportional to this one will also produce approimatel the same unit length on each ais. Some 0 0 calculators have a square feature that automaticall makes the unit length the same on both aes.
5 Applications. Equations of Lines in SlopeIntercept and Standard Form ( 9) 8 The slopeintercept and standard forms are both important in applications. E X A M P L E 7 Changing forms A landscaper has a total of $800 to spend on bushes at $0 each and trees at $0 each. So if is the number of bushes and is the number of trees he can bu, then Write this equation in slopeintercept form. Find and interpret the intercept and the slope. Write in slopeintercept form: The slope is and the intercept is (0, 6). So he can get 6 trees if he bus no bushes and he loses of a tree for each additional bush that he purchases. WARMUPS True or false? Eplain our answer.. There is onl one line with intercept (0, ) and slope.. The equation of the line through (, ) with slope is.. The vertical line has no intercept.. The equation has a graph that is a vertical line.. The line is perpendicular to the line. 6. The line is parallel to the line. 7. The line 8 has a slope of. 8. Ever straight line in the coordinate plane has an equation in standard form. 9. The line is perpendicular to the line. 0. The line has no intercept.. EXERCISES Reading and Writing After reading this section, write out the answers to these questions. Use complete sentences.. What is the slopeintercept form for the equation of a line?. How can ou determine the slope and intercept from the slopeintercept form.. What is the standard form for the equation of a line?. How can ou graph a line when the equation is in slopeintercept form?
6 86 ( 0) Chapter Linear Equations in Two Variables and Their Graphs. What form is used in this section to write an equation of a line from a description of the line? 6. What makes lines look perpendicular on a graph?. Write an equation for each line. Use slopeintercept form if possible. See Eample
7 . Equations of Lines in SlopeIntercept and Standard Form ( ) Draw the graph of each line using its intercept and its slope. See Eamples and Find the slope and intercept for each line that has a slope and intercept. See Eample Write each equation in standard form using onl integers. See Eample
8 88 ( ) Chapter Linear Equations in Two Variables and Their Graphs Write an equation in slopeintercept form, if possible, for each line. See Eample 6. In each case, make a sketch. 6. The line through (0, 6) that is perpendicular to the line 6. The line through (0, ) that is perpendicular to the line 6. The line with intercept (0, ) that is parallel to the line 66. The line through the origin that is parallel to the line The line through (, ) that runs parallel to the ais 68. The line through (, ) that runs parallel to the ais 69. The line through (0, ) and (, 0) The line through (0, ) and (, 0) Solve each problem. See Eample Marginal cost. A manufacturer plans to spend $0,000 on research and development for a new lawn mower and then $00 to manufacture each mower. The formula C 00n 0,000 gives the cost in dollars of n mowers. What is the cost of 000 mowers? What is the cost of 00 mowers? B how much did the one etra lawn mower increase the cost? (The increase in cost is called the marginal cost of the 00st lawn mower.) 7. Marginal revenue. A defense attorne charges her client $000 plus $0 per hour. The formula R 0n 000 gives her revenue in dollars for 6. 0 Revenue (thousands of dollars) Time (hours) FIGURE FOR EXERCISE 7
9 . Equations of Lines in SlopeIntercept and Standard Form ( ) 89 n hours of work. What is her revenue for 00 hours of work? What is her revenue for 0 hours of work? B how much did the one etra hour of work increase the revenue? (The increase in revenue is called the marginal revenue for the 0st hour.) 7. Inhouse training. The accompaning graph shows the percentage of U.S. workers receiving training b their emploers. The percentage went from % in ear 0 (98) to 6% in ear (99). a) Find the slope of this line. b) Write the equation of the line in slopeintercept form. c) Use our equation to predict the percentage that will be receiving training in the ear 000. d) What is the slope of the line? e) What does the slope tell ou? 76. Pens and pencils. A bookstore manager plans to spend $60 on pens at 0 cents each and pencils at 0 cents each. The equation can be used to model this situation. a) What do and represent? b) Graph the equation. 0 Percentage of workers receiving training 0 0 (98) Year (99) FIGURE FOR EXERCISE 7 7. Women and marriage. The percentage of women in the 0 to age group who have never married went from 6% in ear 0 (970) to % in ear 6 (996) (Census Bureau, a) Find the equation of the line through the two points (0, 0.6) and (6, 0.) in slopeintercept form. b) Use our equation to predict what the percentage will be in the ear Pansies and snapdragons. A nurser manager plans to spend $00 on 6packs of pansies at 0 cents per pack and snapdragons at cents per pack. The equation can be used to model this situation. a) What do and represent? c) Write the equation in slopeintercept form. d) What is the slope of the line? e) What does the slope tell ou? GRAPHING CALCULATOR EXERCISES Graph each pair of straight lines on our graphing calculator using a viewing window that makes the lines look perpendicular. Answers ma var , 0 b) Graph the equation , 60 c) Write the equation in slopeintercept form.
Let (x 1, y 1 ) (0, 1) and (x 2, y 2 ) (x, y). x 0. y 1. y 1 2. x x Multiply each side by x. y 1 x. y x 1 Add 1 to each side. SlopeIntercept Form
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