1. Linear Systems

3.  When you have two or more related unknowns, you may be able to represent their relationship with a SYSTEM OF EQUATIONS. This is a set of two or more equations.  Essential Understanding – To solve a system of equations, find a set of values that replace the variables in the equation and make EACH equation true.  A linear system consists of linear equations. A solution of a system is a set of values for the variables that makes all the equations true. We are going to learn how to do this using graphs and tables.

4.  Problem 1 Using a graph or a table to solve a system. What is the solution of the system? -5 -4 -3 -2 0 1 2

6. What kind of lines are these? What key feature is the same for each equation? What key feature is different in each equation?

9.  You can classify a system of two linear equations by the number of solutions.  A consistent system has at least one solution. A consistent system can then be classified as INDEPENDENT and DEPENDENT.  An independent system has ONE solution. A dependent system has INFINITELY many solutions.  An inconsistent system has NO solution.

10.  Problem 4Classifying a system without graphing Without graphing, is the system independent, dependent, or inconsistent?  Got it?Without graphing, is the system independent, dependent, or inconsistent? Inconsistent Slopes = 3 Y-intercepts different Independent Two different lines Dependent Same exact line

11. Page 6 in packet Objective: To solve linear systems algebraically.

12.  When you try to solve a system of equations by graphing, the coordinates of the point of intersection may not be obvious.  If two numbers are substituted for x and y in a system of equations and they make BOTH equations TRUE, then the ordered pair (x, y) is a solution of the system.  Essential Understanding – You can solve a system of equations by writing equivalent systems until the value of one variable is clear. Then substitute to find the value(s) of the other variable(s).  You can use the SUBSTITUTION METHOD to solve a system of equations when it is easy to isolate one of the variables.  After isolating the variable, substitute for that variable in the other equation.  Then solve for the other variable. (Check your solution by plugging back into the original equations.)

13. Problem 1 What is the solution of the system of equations? Step 1Step 2Step 3 Solve one equation for one of the variables Substitute the expression for y in the other equation. Solve for x. Substitute the value for x into one of the original equations. Solve for y.

14.  What is the solution of the system of equations?

15. MusicA music store offers piano lessons at a discount for customers buying new pianos. The costs for lessons and a one-time fee for materials (including music books, CDs, software, etc.) are shown in the advertisement. What is the cost of each lesson and the one-time fee for materials?

16. Let c = the cost of one song Let f = the one-time fee. Got It?An online music company offers 15 downloads for $19.75 and 40 downloads for $43.50. Each price includes the same one-time registration fee. What is the cost of each download and the registration fee? 15 cost of one song + one-time fee = $19.75 40 cost of one song + one-time fee = $43.50

17. Page 9 in your packet Objective To solve systems of Inequalities http://www.phschool.com/atschool/academy123/english/academy123_content/wl-book-demo/ph-238s.html

18.  You can solve a system of inequalities in more than one way. Graphing the solution is usually the most appropriate method. The solution is the set of all points that are solutions of each inequality in the system.  Problem 1Solving a system by using a table Which numbers satisfy BOTH inequalities? gm 0 1 2 3 4 6, 7, 8, 9, 10 5, 6 4, 5 None Are these the only numbers that work as solutions?

19.  What is the solution of the system of inequalities?

20.  What is the solution to the system of inequalities?

22. Fund RaisingYour city’s cultural center is sponsoring a concert to raise at least $30,000 for the city’s Youth Services. Tickets are $20 for BALCONY seats and $30 for ORCHESTRA seats. If the center has 500 orchestra seats, how many of each type of seat much be sold? http://www.phschool.com/atschool/academy123/english/academy123_content/wl-book-demo/ph-512s.html Let x = balcony seats Let y = orchestra seats 20 balcony seats + 30 orchestra seats ≥ $30,000 orchestra seats ≤ 500

23. A pizza parlor charges $1 for each veggie topping and $2 for each meat topping. You are hungry and want at least 5 toppings on your pizza. You have only $10 to spend on the toppings, how many of each type of topping can you get on your pizza? Let x = veggie topping Let y = meat topping $1 veggie topping + $2 meat topping ≤ $10 veggie topping + meat topping ≥ 5

24. What is the solution of the system of inequalities?