#### Transcript Chapter 8: Linear Equations as Models

Chapter 8: Linear Equations as Models Advanced Math Section 8.1: Linear Growth and Decay Slope – Intercept Form of an Equation y = mx +b ; m = slope ; b = y – intercept Slope rise over run change in y over the change in x Without graphing, find the slope and the vertical intercept of the line modeled by each equation. Y = 7x + 4 Y = ½x + 17 Slope = 7 Y – intercept = 4 Slope = ½ y – intercept = 17 Try these on your own… Y = 9 + 4x Slope = 4 Y-intercept = 9 Y = 7/6x + 2 Slope = 7/6 Y-intercept = 2 Sample #1 When you travel up from Earth’s surface, the air temperature decreases by about 11 degrees for each mile you rise about the ground. Suppose the temperature of the air at ground level is 68 degrees. Model the situation with a table of values. Use your table to make a graph. Find the slope and the vertical intercept of the line that contains the points. Explain what they mean in terms of the situation. Write an equation for the temperature of the air (t) as a function of the height in miles above the ground (h). Try this one on your own… Frankie has $500 in her savings account at the beginning of the year. Each month she saves $150. Model the situation with a table of values and a graph. Find the slope and the vertical intercept of the line. Explain what they mean in terms of the situation. Write an equation for the savings (s) as a function of the time in months (t). Linear Growth and Decay Linear Growth Same Value Repeatedly Added Positive Slope Linear Decay Same Value Repeatedly Subtracted Negative Slope Sample #2 Classify each situation as a linear growth, linear decay, or neither. Explain your choice. Eleanor Basave put $5 in a piggy bank for her granddaughter when she was born. Every day after that Eleanor put a quarter in the bank. Linear Growth During each year, the shortest distance between Earth and the sun is 94,000,000 miles and the greatest distance is 94,500,000. Neither A teacher on a remote Arctic island buys 144 cans of fruit for the year. Each week the teacher eats 4 cans of fruit. Linear Decay Section 8.2: Linear Combinations A solution of an equation with two variables is an ordered pair of numbers that makes the equation true. All the points whose coordinates are solutions of an equation form the graph of the equation. Standard Form of an Equation for a Line The equation 8t + 12d = 48 is called a linear equation because it is an equation of a line. Here is the standard form of a linear equation… ax + by = c Sample #1 Several times a week, Dynah Colwin runs part of the way and walks part of the way on a trail that is 4 miles long. Her running speed is 6 miles per hour and her walking speed is 4 miles per hour. Write an equation for this situation. Let r = the time in hours Dynah Colwin spends running Let w = the time in hours she spends walking Rewrite the equation in slope-intercept form. Try this one on your own… Each week, Will swims the backstroke part of the way and the crawl part of the way for 30 laps of the swimming pool. His backstroke speed is 3 laps per minute and his crawl speed is 2 laps per minute. Write an equation for this situation. Let b = time it takes to swim the backstroke Let c = time it takes to swim the crawl Rewrite the equation in slope-intercept form. Intercepts of a Line Vertical Intercept (y – intercept) x=0 Horizontal Intercept ( x – intercept) y=0 Sample #2 Find the intercepts of the graph 2x – 5y = 15. Use them to graph the equation. Try this one on your own… Find the intercepts of the graph of y = -2x + 2. Use them to graph the equation. Section 8.3: Horizontal and Vertical Lines When you have a horizontal line, the slope is zero. The coefficient of x will be zero. When you have a vertical line, the slope is undefined. The coefficient of y will be zero. Sample #1 Find the slope of each line and write an equation for each line. A The slope is 0. The y-coordinate is always 2.5. Y = 2.5 B The slope is undefined. The x-coordinate is always -1. X = -1 Try these on your own… Find the slope of each line and write an equation for each line. A Slope = 0 y = -7.6 B Slope = undefined x = 1.2 Sample #2 Write an equation for each line. The point (-2,4) and (-2,0) are on the line. X-coordinates are both -2. Every point on the line has the same x coordinate, -2. X = -2 The slope is zero, and the point (2,3) is on the line. Slope is zero. Every point on the line has the same y coordinate, 3 Y = 3 Try these on your own… Write an equation for each line. The point (9,3) and (-1,3) are on the line. y=3 The slope is undefined, and the point (-1,4) is on the line. x = -1 Section 8.4: Writing Equations for Lines Estimate an equation for the fitted line for the Olympic data. Step One: Find the slope. Step Two: Find the yintercept. Step Three: Re-write equation (y = mx + b) filling in the slope and y-intercept. Try this one on your own… Estimate an equation for the fitted line for the data. Interpret Week 0 to mean the beginning of the first week. Slope = 4 y – intercept = 15 y = 4x + 15 Sample #2: Exact Equation from Two Points Two points on a line are (4,3) and (-6,-2). Write an equation for the line. Step One: Find the slope. Step Two: Find the y-intercept. Step Three: Re-write equation (y = mx +b) inputting the slope and y-intercept. Try this one on your own… Two points on a line are (1, -2) and (5,4). Write an equation for the line. Step One: Find the slope. Step Two: Find the y-intercept. slope = 3/2 y – intercept = -7/2 Step Three: Re-write equation. y = 3/2x – 7/2 Sample #3: Equations from Other Facts The slope of a line is 3/2. One point on the line is (-4,1). Write an equation for the line. Two Methods to Find the Y-Intercept Substitution Use the Slope-Intercept Formula Graph Try this one on your own… The slope of a line is -1/2. One point on the line is (3,4). Write an equation for the line. Section 8.5: Graphing Systems of Linear Equations MoviesPlus rents videos for $2.50 each and has no membership fee. Videobusters rents videos for $2 each but has a $10 membership fee. Write and solve a system of equation to model this situation. (Two Options – Graphing and Setting Them Equal) Let c = total cost of renting videos Let n = number of videos rented What advice would you give to someone trying to decide which video store to use? Try this one on your own… The service charge on a checking account at Hometown Bank is $5 per month plus $0.15 for each check written. The service charge at Twentieth Century Bank is $0.25 per check. Write and solve system of equations to model this situation. Let c = cost of the service charge Let n = number of checks written What advice would you give to someone trying to decide which bank to use? Sample #2: Systems without Solutions Sandy and Rita are practicing for 100 m dash competition. Sandy gives Rita a 10 m head start. Write and graph a system of equations to model this situation. Let x = time since the runner started Let y = runner’s distance from starting line Try this one on your own… James left the campground riding his bicycle at 15 miles per hour. When he was 5 miles away from camp, Lucie left the camp riding at the same speed. Write and graph a system of equations to model this situation. Let x = time Lucie started Let y = their distance from the camp Section 8.6: Graphing Linear Inequalities Linear Inequalities on a Coordinate Plane The graph of a linear inequality is a region on a coordinate plane whose edge is a line. The line is called a boundary line. Sample #1: Graph each inequality. yx 1 y x4 2 Try these on your own… y x 1 y x 1 2 Talk it Over… Tell whether the boundary line for each inequality is solid or dashed. Then tell whether you would shade the region above or below the line. y 9x 6 2 y x 3 y 4 x 11 1 y x 4 Sample #2 Derek has saved $48 by baby-sitting. He plans to use the money to buy tapes and compact discs at a music store. Tapes cost $8 and compact discs cost $12. The inequality 8x 12y 48 represents the amount Derek can spend at the store. Graph the inequality 8x 12y 48 Try this one on your own… Graph the inequality 5y – 3x > 15. Dashed or Solid Line? Dashed Shade Above or Below? Below Sample #3: Graphing Inequalities in Standard Form Graph the inequality 8x – 12y < 48. Use the intercepts to graph the boundary line. Dashed or Solid? Check test point. Try this one on your own… Graph the inequality 5 y 3x 15