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Essential Question:

Learn how to write and solve real world application problems using slope-intercept form and standard form of linear equations. Explore examples and find the cost with given information.

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Essential Question:

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  1. Essential Question: How can I write and solve real world application problems using slope Adapted by Christopher Carnes, RVMS, Hemet, CA

  2. Real World Applications Slope-Intercept Form Standard Form

  3. Slope-Intercept Form Word problems with linear equations will be in slope-intercept form if you are given the following information: • the slope (m) - the rate of change • the y-intercept (b) - the starting (initial) value

  4. for each hour $30 plus Cost is $100 + = y 30 100 •x Application A closet organizer charges a $100 initial consultation fee plus $30 per hour. The cost as a function of the number of hours worked is graphed below. a. Write an equation that represents the cost as a function of the number of hours. An equation is y = 30x + 100.

  5. Continued A closet organizer charges $100 initial consultation fee plus $30 per hour. The cost as a function of the number of hours worked is graphed below. b. Identify the slope and y-intercept and describe their meanings. The y-intercept is 100. This is the cost for 0 hours, or the initial fee of $100. The slope is 30. This is the rate of change of the cost: $30 per hour. c. Find the cost if the organizer works 12 hrs. y = 30x + 100 Substitute 12 for x in the equation = 30(12) + 100 = 460 The cost of the organizer for 12 hours is $460.

  6. for each meal $18 plus Cost is $200 200 + = •x y 18 Try This! A caterer charges a $200 fee plus $18 per person served. The cost as a function of the number of guests is shown in the graph. a. Write an equation that represents the cost as a function of the number of guests. An equation is y = 18x + 200.

  7. Try This! Continued A caterer charges a $200 fee plus $18 per person served. The cost as a function of the number of guests is shown in the graph. b. Identify the slope and y-intercept and describe their meanings. The y-intercept is 200. This is the cost for 0 people, or the initial fee of $200. The slope is 18. This is the rate of change of the cost: $18 per person. c. Find the cost of catering an event for 200 guests. y = 18x + 200 Substitute 200 for x in the equation = 18(200) + 200 = 3800 The cost of catering for 200 people is $3800.

  8. More – finding slope with formula You will start your linear equation by finding slope – how many points do you need to find slope??? If you are given the following type of information: • A table of information • Information that can be written as ordered pairs

  9. Problem-Solving Application The cost to stain a deck is a linear function of the deck’s area. The cost to stain 100, 250, 400 square feet are shown in the table. Write an equation in slope-intercept form that represents the function. Then find the cost to stain a deck whose area is 75 square feet. • The answer will have two parts—an equation in slope-intercept form and the cost to stain an area of 75 square feet. • The ordered pairs given in the table—(100, 150), (250, 337.50), (400, 525)—satisfy the equation.

  10. Continued You can use two of the ordered pairs to find the slope. Then use point-slope form to write the equation. Finally, write the equation in slope-intercept form. Choose any two ordered pairs from the table to find the slope. Use (100, 150) and (400, 525). Substitute the slope and any ordered pair from the table into y = mx + b

  11. Continued Now we can solve for the y-intercept. Write the equation and use any point. y = 1.25x + b Write equation without b 150 = 1.25(100) + b Substitute one point for (x,y) 150 = 125 + b Simplify 25 = b Subtract 125 from each side Substitute the y-intercept back into the equation y = 1.25x + 25 y = 1.25(75) + 25 = 118.75 The cost of staining 75 sq. ft. is $118.75.

  12. y = 1.25x + 25 y = 1.25x + 25 y = 1.25x + 25 525 1.25(400) + 25 337.50 1.25(250) + 25 525 500 + 25 337.50 312.50 + 25  525 525  337.50 337.50 Continued If the equation is correct, the ordered pairs that you did not use in Step 2 will be solutions. Substitute (400, 525) and (250, 337.50) into the equation.

  13. Try This! What if…? At a newspaper the costs to place an ad for one week are shown. Write an equation in slope-intercept form that represents this linear function. Then find the cost of an ad that is 21 lines long. • The answer will have two parts—an equation in slope-intercept form and the cost to run an ad that is 21 lines long. • The ordered pairs given in the table—(3, 12.75), (5, 17.25),(10, 28.50)—satisfy the equation.

  14. Try This! Continued You can use two of the ordered pairs to find the slope. Then use y = mx + b to find b Step 1 Choose any two ordered pairs from the table to find the slope. Use (3, 12.75) and (5, 17.25). Step 2 Substitute the slope and any ordered pair from the table

  15. Try This! Continued Now, lets solve for the y-intercept: y = 2.25x + b Write the equation without b 12.75 = 2.25(3) + b Substitute one point and solve for b 12.75 = 6.75 + b Simplify 6 = b Subtract 6.75 from both sides Substitute the y-intercept back into the equation. y = 2.25x + 6 y = 2.25(21) + 6 = 53.25 The cost of the ad 21 lines long is $53.25.

  16. y= 2.25x + 6 y = 2.25x + 6 12.75 2.25(3) + 6 28.50 2.25(10) + 6 12.75 6.75 + 6 28.50 22.50 + 6   12.75 12.75 28.50 28.50 4 Look Back Try This! Continued If the equation is correct, the ordered pairs that you did not use in Step 2 will be solutions. Substitute (3, 12.75) and (10, 28.50) into the equation.

  17. Standard Form You will write an equation in standard form if you are given the following type of information: • Values for two types of items • The values are combined to equal one total.

  18. 2x + 3y = 60 2x + 3y = 60 0 + 3y = 60 2x + 0 = 60 Application The school sells pens for $2.00 and notebooks for $3.00. The equation 2x + 3y = 60 describes the number of pens x and notebooks y that you can buy for $60. Graph the function and find its intercepts. 2x + 3y = 60 Find the intercepts. x-intercept: y-intercept: 3y = 60 2x = 60 x = 30 y = 20 x-intercept: 30; y-intercept: 20

  19. Application The school sells pens for $2.00 and notebooks for $3.00. The equation 2x + 3y = 60 describes the number of pens x and notebooks y that you can buy for $60. What does each intercept represent? x-intercept: 30. This is the number of pens that can be purchased if no notebooks are purchased. y-intercept: 20. This is the number of notebooks that can be purchased if no pens are purchased.

  20. Try This! You are buying $20 worth of bird seed that consists of two types of seed. Thistle seed costs $2 per pound and Dark Oil Sunflower Seed costs $5 per pound. Write an equation that represents the different amounts of each type of seed you can buy. Graph this function by finding the intercepts. What does each intercept represent? The equation of the line will be: 2x + 5y = 20 x-intercept: y-intercept: 2x = 20 x = 10 5y = 20 y = 4 x-intercept: 10. This is the amount of Thistle Seed that can be purchased if no Sunflower Seed is purchased. y-intercept: 4. This is the amount of Sunflower Seed that can be purchased if no Thistle Seed is purchased.

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