Alg1 10212016

Warm Up The Pendleton County School District is trying to decide on a new copier. The purchasing committee has been given quotes on two new machines. One sells for $20,000 and costs $0.02 per copy to operate. The other sells for $17,500, but its operating cost is $0.025 per copy. Your teacher will give you extra credit for writing a mathematically sound recommendation to the school district on which copier to buy based on the total cost during the first year. 1. The district estimates the number of copies made each year is 515,000. Based on this estimation, which machine would you recommend? Justify your choice with clear mathematics.

Warm Up-debrief The district estimates the number of copies made each year is 515,000. Based on this estimation, which machine would you recommend? Justify your choice with clear mathematics. The total cost for each copier can be determined using equations. Let C represent the operating cost and p represent the number of pages copied. Copier 1: C = 20,000 + 0.02p Copier 2: C = 17,500 + 0.025p Substitute 515,000 for the number of copies, p, to determine the cost for each copier. Copier 1: C = 20,000 + 0.02(515,000) = 30,300 Copier 2: C = 17,500 + 0.025(515,000) = 30,375 Interpret Copier 1 costs $30,300 for 515,000 copies. Copier 2 costs $30,375 for 515,000 copies. Copier 1 would be the best copier for this district because it is cheaper at the estimated number of pages.

Standard A–REI.11 Explain why the x-coordinates of the points where the graphs of the equations y=f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.★

Learning Targets Solve a system of equations using various methods and justify thinking. Take a scenario a step further and determine at which point the equations are equal .

Mini Lesson Two or more equations that are solved together are called systems of equations. The solution to a system of equations is the point or points that make both equations true. Systems of equations can have one solution, no solutions, or an infinite number of solutions.

Mini Lesson Finding the solution to a system of equations is important to many real-world applications. There are various methods to solving a system of equations. From a graph From a table Algebraically Today we will explore Algebraically looking at two methods: the substitution method and the elimination method.

Mini Lesson The substitution method involves solving one of the equations for one of the variables and substituting that into the other equation.

Mini Lesson Substitution method A solution to a system of equations is written as an ordered pair, (x, y). This is the point where the lines would intersect if graphed. If the resulting solution is a true statement, such as 9 = 9, then the system has an infinite number of solutions. The lines would coincide if graphed. If the result is an untrue statement, such as 4 = 9, then the system has no solutions. The lines would never intersect, so they would be parallel if graphed. Check your answer by substituting the x and y values back into the original equations. If the answer is correct, the equations will result in true statements.

Mini Lesson Solving Systems of Linear Equations by Elimination Using Addition or Subtraction • The elimination method involves adding or subtracting the equations in the system, using properties of equality, so that one of the variables is eliminated.

Mini Lesson Solving Systems of Linear Equations by Elimination Using Multiplication • Use this method when the coefficients of one of the variables are neither opposites nor the same. The multiplication property of equality can be used with one or both equations in order to make one pair of coefficients match.

Practice

Practice

Practice

Practice Practice 2.3.1: Intersecting Graphs Solving using TABLES #5, 6, and 8 ONLY BE SURE TO HAVE WORKBOOKS TOMORROW!!!!

Closing-From Tuesday Brian is 2 years older than his brother Andrew. Brian likes to remind Andrew that he is both older and taller and tells Andrew that it will always be this way. But one day their father overhears Brian and makes a prediction. “Andrew, your brother will always be older than you, but one day you will be taller—and I can prove it mathematically.” He writes two equations, one describing Brian’s growth rate, the other describing Andrew’s growth rate. Both equations are in terms of Brian’s age in years, x, and each boy’s height, y, in feet. Assume that the boys will finish growing at age 18. Use the equations to complete the problems. Brian’s growth rate: B(x)=1/5x+2.25 Andrew’s growth rate: A(x)=1/4(x−2)+2 3. Will Andrew ever be taller than Brian? Explain your answer. 4. If Andrew does outgrow Brian, how old will Andrew be when he does?

Homework Check - Practice 2.3.1: Intersecting Graphs Use what you know about graphing functions to complete each problem. Given the graphs of y=f(x) and y=g(x), what is the value of f(x)-g(x) where the two graphs intersect? If you are using the graphs of y=f(x) and y=g(x) to estimate x where f(x)=g(x), and you see that the graphs intersect at two different points, how many values of x should you attempt to find? 2