1. Slide 1 Lesson 77 Offsetting Rates CP.4 Determine the offset for non-proportional relationships involving rates or ratios and represent them with lines that do not pass through the origin. CP.5 Know that the graph of an equation is the set of all points which are solutions to the equation. CP.2 Plot solutions to linear equations on the coordinate plane. CP.14 Graph linear equations. Chapter 14 Lesson 77 CP.1 Identify and plot ordered pairs on the coordinate plane.

2. Slide 2 Lesson 77 Objectives Write and solve two-step linear equations. Graph linear equations that are offset from the origin.

3. Slide 3 Lesson 77 Remember from Before How does a point on the coordinate plane represent a solution to an equation? What is a graph of an equation? What is a linear equation?

4. Slide 4 Lesson 77 Get Your Brain in Gear 1. Use mental math to find 5 solutions to each equation.

5. Slide 5 Lesson 77 1 pineapple costs 2 dollars. The following expression tells us how many pineapples we can buy for d dollars: We can use the variable p to represent the output number of pineapples. Here are some solutions to this equation: If we could plot all of the solutions to this equation, we would get a straight line.

6. Slide 6 Lesson 77 What if the store gives away 1 free pineapple to anyone that comes to the store? This means that for any number of dollars we spend, we will get 1 more pineapple for free. Use the variable p to represent the total number of pineapples. Every input results in 1 extra pineapple for the output.

7. Slide 7 Lesson 77 Each output is now 1 more than before. On the coordinate grid this means that each point will get offset upward by 1 pineapple. When we multiply an input by a rate and then add a constant value to get an output, the graph will be a straight line. If the constant isn’t zero, then the line won’t go through the origin.

8. Slide 8 Lesson 77 Check for Understanding 1. Match each of these situations to one of the graphs that best represents the situation. Explain your reasoning. Situation 1: There are 3 pens in every box. If you have b boxes, how many pens do you have? Situation 2: There are 3 pens in every box. If you have b boxes and 2 extra pens, how many pens do you have? Situation 3: There are 3 pens in every box. If you have b boxes and give away 2 of your pens, how many pens will you have?

9. Slide 9 Lesson 77 Check for Understanding 2. Match each equation to a situation below. Situation 1: There are 3 pens in every box. If you have b boxes, how many pens do you have? Situation 2: There are 3 pens in every box. If you have b boxes and 2 extra pens, how many pens do you have? Situation 3: There are 3 pens in every box. If you have b boxes and give away 2 of your pens, how many pens will you have?

10. Slide 10 Lesson 77 Outside there is a big pile of dirt that is 8 feet tall: If the pile gets 2 feet shorter every 3 weeks, how tall will it be after w weeks? w represents the number of weeks.When w = 0, the above expression equals 0 feet. However, the problem states that the pile starts out 8 feet tall. This means we need to add 8 feet to the above expression. What is the rate? Since the pile decreases in height, we represent the change in height as a negative number. Let’s use the variable f to represent how many feet tall the pile is.

11. Slide 11 Lesson 77 Simplify: Use the distributive property: This gives us:Simplify:

12. Slide 12 Lesson 77 Let’s now find some solutions to this equation. How tall will the pile of dirt be after 0 weeks? This solution means that the pile of dirt is 8 feet tall after zero weeks.

13. Slide 13 Lesson 77 What is the solution when w equals 3? Simplify: w = 3 and f = 6 is another solution:

14. Slide 14 Lesson 77 Every 3 weeks the pile will be 2 feet shorter:

15. Slide 15 Lesson 77 What will the solutions look like on the coordinate plane? Since we are offsetting a constant rate by a constant value, the graph of this equation is a straight line.

16. Slide 16 Lesson 77 Notice that after 12 weeks, the height of the dirt is a negative value. What does this mean? What is f when w equals 15? Simplify: w = 15 and f = _ 2 is a solution:

17. Slide 17 Lesson 77 What does _ 2 feet mean? We know that +2 feet means that the height of the dirt is 2 feet above the ground. The opposite of being above the ground is being below the ground. A height of _ 2 feet is a hole that is 2 feet deep into the ground. After 15 weeks, the pile of dirt went from 8 feet tall down to a hole that is 2 feet into the ground:

18. Slide 18 Lesson 77 We can plot this solution on the coordinate plane by placing a point at (15, _ 2): Notice that this point is on the same line as all the other solutions.

19. Slide 19 Lesson 77 Check for Understanding 3. Consider the following equation where the input is k kicks and the output is y yards: a.What is the rate and what is the offset? b. Simplify the units in the above equation. c.Find 3 solutions to the equation. d. Plot the solutions on a coordinate plane. e. Fit a line through the plotted points. f.What input gives an output of zero yards? g.What is the output when the input is 0 kicks?

20. Slide 20 Lesson 77 1. Which equation below has a graph that goes through the origin? Multiple Choice Practice

21. Slide 21 Lesson 77 2. Which graph most reasonably represents the following equation? Multiple Choice Practice

22. Slide 22 Lesson 77 Find the Errors A student made an error in the following equation. Identify and correct the error.