I(g) = income from selling gearboxes C(g) = cost of purchasing gearboxes The BREAK-EVEN PT is where COST = INCOME or C(g) = I(g).

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2 I(g) = income from selling gearboxes C(g) = cost of purchasing gearboxes The BREAK-EVEN PT is where COST = INCOME or C(g) = I(g). PROFIT is when INCOME > COST or I(g) > C(g).

3 I(g) = 8.5g g = the number of gear boxes C(g) = 5.77g + 45 I(g) = 8.5g C(g) = 5.77g + 45 Break-even Point = Point of Intersection a. How is the break-even point for I(g) and C(g) represented on the graph you sketched? Estimate the break-even point. The break-even point is between 15 and 20 gearboxes near the point (15, 125). So, RR must sell more than 15 but less than 20 gearboxes to break-even.

4 b. Could you determine the exact break-even point from the graph? Why or why not. No, because the two function graphs or lines do not intersect at an exact location on the x- and y-axes. As you learned previously, the coordinates of an intersection of two graphs can be exact or approximate depending on whether the intersection point is located on the intersection of two grid lines. You also learned that you had to use algebra to prove an exact intersection point. When determining the break-even point algebraically between two functions, it is more efficient to transform each function into equation form. In this case, by transforming the functions into equation form, you establish one unit of measure for the dependent quantity: dollars.

5 4. Do you think it is possible to use other variables instead of x and y when transforming a function written in function notation to equation form? Yes. You can use any variables you want as long as you choose the same variables to represent the independent and dependent quantities in each equation. When two or more equations define a relationship between quantities, they form a system of linear equations. 5. What is the relationship between the two equations in this problem situation? One equation represents the COST (in dollars) of buying the gearboxes, and the other equation represents the INCOME (in dollars) earned by selling the gearboxes.

6 Now that you have successfully created a system of linear equations, you can determine the break-even point for the gearboxes at RR. One way to solve a system of linear equations is called the substitution method. The substitution method is a process of solving a system of equations by substituting a variable in one equation with an equivalent expression. ***Important!!!

7 6. Analyze the solution x a. What does this point represent in terms of the problem situation? Why is this solution an approximation? This point represents the # of gearboxes RR must sell before the company starts making money. It is an approximation because the solution is rounded to the 100ths place. b. Solve for y. Describe the solution in terms of this problem situation. y = 8.5x y = 8.5(16.48) Plug in the value for x to solve for y. y This point represents the money RR paid to purchase these gearboxes (COST) as well as the amount of money they made selling these gearboxes (INCOME). c. What is the profit from gearboxes at the break-even point? The profit is $0. d. Does this break-even point make sense in terms of the problem situation? Why or why not. No. RR cannot sell gearboxes. They will have to sell 17 gearboxes.

8 7. Analyze your graph of the cost and the income for the different number of gearboxes. a. Draw a box around the portion of the graph that represents when RR is losing money. Then write an inequality to represent this portion of the graph and describe what it means in terms of the problem situation. RR loses money when x < 16.48, or they sell fewer than 17 gearboxes. COST > INCOME b. Draw an oval around the portion of the graph that represents when RR is earning money. Then write an inequality to represent this portion of the graph and describe what it means in terms of the problem situation. RR earns money or makes a profit when x > or they sell 17 or more gearboxes. INCOME > COST c. Write an equation to represent the portion of the graph that represents when RR breaks even and describe what it means in terms of the problem situation. RR breaks even when x = Since RR cannot sell gearboxes, there is no EXACT point when RR will break even.

9 Yesterday we graphed the Income function (from selling gearboxes) and the Cost function (associated with purchasing the gearboxes). 1. What does the intersection point of the two functions represent? The break-even point 2. What does the area where the oval is drawn represent? Where Income is greater than cost 3. What does the area where the box is drawn represent? Where cost is greater than income

10 Page 372 w = time (in weeks) M(w) = amount of money (in dollars) Marcus saves P(w) = amount of money (in dollars) Phillip saves M(w) = P(w) M(w) = w P(w) = w They will never have the same amount of money saved because Phillip starts with $40 and Marcus starts with $25.

11 You can prove your prediction by solving and graphing a system of linear equations. 4. Rewrite each function as an equation. Use x and y for the variables of each function in equation form and define the variables. Then, write a system of linear equations. x = time (in weeks) y = the amount of money saved (in dollars) y = x y = x 5. Analyze each equation. a. Describe what the slope of each line represents in this problem situation. The slope represents the amount of money each person saves each week. b. How do the slopes compare? Describe what this means in terms of this problem situation. The slopes are the same because both people save the same amount of money each week, $10. c. Describe what the y-intercept of each line represents in this problem situation. The y-intercept represents the amount of money each friend started with. d. How do the y-intercepts compare? Describe what this means in terms of this problem situation. Phillip s y-intercept is greater than Marcus s y-intercept because Phillip started with more money than Marcus.

12 6. Determine the solution of the system of linear equations algebraically and graphically. a. Use the substitution method to determine the intersection point. y = x y = x 25 10x 40 10x 10x 10x b. Does your solution make sense? Describe what this means in terms of the problem situation. No, 25 does not equal 40 so there is no solution. c. Predict what the graph of this system will look like. Explain your reasoning. The linear equations have the SAME SLOPE, but DIFFERENT Y-INTERCEPTS so they must be parallel.

13 d. Graph both equations on the coordinate plane provided. 7. Analyze the graph you created. a. Describe the relationship between the graphs. The graphs are the same distance apart from one another which means they are parallel. b. Does this linear system have a solution? Explain your reasoning. No. The graphs will never intersect because they are parallel, so there is no solution. 8. Was your prediction in Question 3 correct? Explain how you algebraically and graphically proved your prediction. Yes. Using the substitution method, my solution of proved that there is no intersection point. My graph shows two parallel lines which will never intersect and proves there is no intersection point. Skip #9 and #10

14 Page Phillip and Tonya went on a shopping spree this weekend and spent all their savings except for $40 each. Phillip is still saving $10 a week from his allowance. Tonya now deposits her tips twice a week. On Tuesdays she deposits $4 and on Saturdays she deposits $6. Phillip claims he is still saving more each week than Tonya. a. Do you think Phillip s claim is true? Explain your reasoning. No. Phillip and Tonya are now saving the same amount each week and they have the same amount of money left in their accounts. b. How can you prove your prediction? By writing a new system of linear equations and solving them algebraically and graphically. 12. Prove your prediction algebraically and graphically. a. Write functions that represent any new information about the way Tonya and Phillip are now saving money. T ( w) 40 6w 4w P( w) 40 10w b. Write a new linear system to represent the total amount of money each friend has after a certain amount of time. y 40 6x 4x y 40 10x

15 c. Graph the linear system on the coordinate plane. 13. Analyze the graph. a. Describe the relationship between the graphs. What does this mean in terms of this problem situation? The graphs are the same. This means that Tonya and Phillip will always have the same amount of money x 40 6x 4x 40 10x 40 10x b. Algebraically prove the relationship you stated in part (a) c. Does this solution prove the relationship? Explain your reasoning. Yes. Determining that 40 = 40 means there are an infinite number of solutions. Therefore, this means the graphs must be the same. 14. Was Phillip s claim that he is still saving more than Tonya a true statement? Explain why or why not. No. Phillip is not saving more than Tonya. He is saving the same amount as Tonya.

16 a. Describe the method Dontrell used to solve this system of equations and explain why he is correct. Dontrell solved the second equation for y. Then he substituted it in as the y-value in the first equation.

17 b. Describe the method Janelle used to solve this system of equations and explain why her reasoning is correct. Janelle solved for the x-value in the first equation. She then substituted it in for the x- value in the second equation. c. Describe the method Maria used to solve this system of equations and explain why her reasoning is correct. Maria solved for the y-value in both equations. Once she had solved for the y-value for each equation, she set the two equations equal to each other. 2. Which method do you prefer for solving this system of equations? Answers may vary. I prefer Maria s method. I think it is easier to keep track of what I am doing if I solve each equation for the same variable and then set them equal to each other.

18 3. Use one of the methods shown or use your own method to determine the solution to this system of equations.

19 Sammy is correct. By multiplying both equations by 10, all the decimals become whole numbers. However, the solution for x and y will be the same. Explain the mistake(s) Soo Jin made and then determine the correct way to rewrite this system. Soo Jin did not multiply the entire first equation through by 10. If she had done it correctly the first equation would be 35x + 12y = 80. The second equation is correct. The correct way to write this system is:

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