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Introduction Constraint equations are used to model real-life situations that involve limits. Unlike other types of equations used for modeling, constraint.

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Presentation on theme: "Introduction Constraint equations are used to model real-life situations that involve limits. Unlike other types of equations used for modeling, constraint."— Presentation transcript:

1 Introduction Constraint equations are used to model real-life situations that involve limits. Unlike other types of equations used for modeling, constraint equations account for restrictions that exist in a situation. For example, when projecting budgets in business, conditions or restrictions often exist on available time, space, and money. 1 6.3.1: Creating Constraint Equations

2 Introduction, continued When graphing a linear constraint equation, acceptable solutions will appear on the graphed line. Since a line has an infinite number of points, an infinite number of solutions work for that equation. A constraint equation shows what restrictions must be placed on the variables to identify solutions that reflect the real-world limits of the situation. 2 6.3.1: Creating Constraint Equations

3 Key Concepts A constraint equation is an equation that expresses a restriction in the relationship between two quantities. A linear constraint equation is most often given in the form ax + by = c, where a, b, and c are constants, but can be rewritten in slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept. Also, there is a ratio between the two quantities x and y, with a and b serving as conditions or restrictions set on x and y. A ratio is the relationship between two quantities, and can be expressed in words, fractions, decimals, or as a percentage. 3 6.3.1: Creating Constraint Equations

4 Key Concepts, continued The graph of a linear constraint equation is a line whose points represent the feasible solutions (or reasonable solutions) of the equation. The acceptable solutions for a linear equation are given in the form (x, y). Recall that the domain is the set of all x-values that are valid for a given situation’s equation. A typical line graph for a constraint equation would show an x-intercept at (x, 0) if the line crosses the x-axis. Similarly, the line for a constraint equation would show a y-intercept at (0, y) if the line crosses the y-axis. 4 6.3.1: Creating Constraint Equations

5 Common Errors/Misconceptions incorrectly setting up the equations to represent the constrained situations solving for only one variable in a constraint equation misinterpreting the solution for a constraint equation 5 6.3.1: Creating Constraint Equations

6 Guided Practice Example 2 Shawn belongs to a book rental club. His mother gave him a $45 gift card for the club as a birthday present. The club charges members $2.00 to rent e-books and $3.50 for print books. Use a constraint equation to find out how many of each type of book Shawn can rent with his $45 gift card. Find two possible combinations of rentals that would serve as solutions to the equation. Write a general description of other combinations based on the constraint equation. 6 6.3.1: Creating Constraint Equations

7 Guided Practice: Example 2, continued 1.Write a constraint equation for the relationship described. First, define the variables for each type of book rental. Let x be the number of e-books rented and y be the number of print books rented. The cost to rent an e-book is $2.00, or 2x, and the cost to rent a printed book is $3.50, or 3.5y. The gift card limit is $45. Therefore, the equation is 2x + 3.5y = 45. 7 6.3.1: Creating Constraint Equations

8 Guided Practice: Example 2, continued 2.Use the constraint equation to determine two possible combinations of rentals, then interpret the solutions in the context of the problem. Choose a value for x, substitute it into the equation, and then solve for y. 8 6.3.1: Creating Constraint Equations

9 Guided Practice: Example 2, continued Let x = 1. 9 6.3.1: Creating Constraint Equations 2x + 3.5y = 45Constraint equation 2(1) + 3.5y = 45Substitute 1 for x. 2 + 3.5y = 45Multiply. 3.5y = 43Subtract. y = 12.29Divide.

10 Guided Practice: Example 2, continued Round down to the nearest whole number, since it is not possible to rent only part of a book. Therefore, one possible solution for the number of rentals is (1, 12). This means Shawn can rent 1 e-book and 12 print books. He would have money left on the gift card, but not enough to rent a full book. 10 6.3.1: Creating Constraint Equations

11 Guided Practice: Example 2, continued Repeat this process with a new value for x. Let x = 10. 11 6.3.1: Creating Constraint Equations 2x + 3.5y = 45Constraint equation 2(10) + 3.5y = 45Substitute 10 for x. 20 + 3.5y = 45Multiply. 3.5y = 25Subtract. y = 7.14Divide.

12 Guided Practice: Example 2, continued After rounding down to the nearest whole number, another possible solution for the number of rentals is (10, 7). Shawn can rent 10 e-books and 7 print books and still have some money left on the gift card. 12 6.3.1: Creating Constraint Equations

13 Guided Practice: Example 2, continued 3.Describe other combinations based on the constraint equation. In general, any integer point (x, y) below the line 2x + 3.5y = 45 would be a feasible combination of printed books and e-books Shawn can rent with his gift card. 13 6.3.1: Creating Constraint Equations ✔

14 Guided Practice: Example 2, continued 14 6.3.1: Creating Constraint Equations

15 Guided Practice Example 3 Ann is trying to decide whether to sign up for a new “pay as you go” cell phone plan. The plan charges $0.25 per minute for calls and $0.10 per text message, which includes all taxes and fees. She wants to spend no more than $100 monthly for cell service. Write the constraint equation and then graph the solution. Finally, determine the domain of the graph (minutes for calls) and interpret how it relates to this situation. 15 6.3.1: Creating Constraint Equations

16 Guided Practice: Example 3, continued 1.Write a constraint equation for the relationship described. Let x be the number of minutes of calls made in a month, and let y be the number of texts sent in a month. Calls cost $0.25 per minute, or 0.25x, and texts cost $0.10, or 0.1y. Ann’s total monthly budget is $100. Therefore, the equation is 0.25x + 0.1y = 100. 16 6.3.1: Creating Constraint Equations

17 Guided Practice: Example 3, continued 2.Rewrite the equation in slope-intercept form, y = mx + b, to determine the slope and y-intercept. To graph the equation, first rewrite it in slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept. 17 6.3.1: Creating Constraint Equations

18 Guided Practice: Example 3, continued Solve 0.25x + 0.1y = 100 for y. The equation 0.25x + 0.1y = 100 rewritten in slope-intercept form is y = –2.5x + 1000, where –2.5 is the slope and 1,000 is the y-intercept. 18 6.3.1: Creating Constraint Equations 0.25x + 0.1y = 100Original equation 0.1y = –0.25x + 100 Subtract 0.25x from both sides. Divide both sides by 0.1. y = –2.5x + 1000Simplify.

19 Guided Practice: Example 3, continued 3.Graph the result by hand or using a graphing calculator. The graph of y = –2.5x + 1000 is shown on the next slide. 19 6.3.1: Creating Constraint Equations

20 Guided Practice: Example 3, continued 20 6.3.1: Creating Constraint Equations

21 Guided Practice: Example 3, continued 4.Determine the domain of the graph and how it relates to this situation. The graph shows that the domain (x-values) is all real numbers. However, x, which represents voice calls at $0.25 per minute, must be less than or equal to 400, since that is the total number of minutes allowed for voice calls if no text messages are sent. 21 6.3.1: Creating Constraint Equations

22 Guided Practice: Example 3, continued The least number of voice calls that could be made is 0. Therefore, for the real-world situation, the restricted domain would be all real numbers such that 0 ≤ x ≤ 400. 22 6.3.1: Creating Constraint Equations ✔

23 Guided Practice: Example 3, continued 23 6.3.1: Creating Constraint Equations

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