MJ2A

1. MJ2A Ch 8.9 – System of Equations

2. Bellwork • Solve for y, given the value of x • y = x + 1 x = 2 • y = x – 4 x = -1 • x + y = -1 x = 1 • x + y = 0 x = 4 Solutions 3 -1 -2 -4

3. Assignment Review • Text p. 407 – 408 # 15 – 25

4. Before We Begin… • Please take out your notebook and get ready to work… • In the last lesson we worked with writing linear equations using the slope-intercept form. • In today’s lesson we will look at pairs of linear equations and their common solutions…

5. Objective 8.9 • Students will solve systems of linear equations by the graphing & substitution methods

6. Vocabulary • Systems of Equations are sets of linear equations. • Solution – is the common ordered pair that make the system of equations true.

7. System of Equations • There are several methods to solving systems of equations. • In today’s lesson we will look at • Solving by graphing • Solving by substitution

8. Solving by Graphing • When solving systems of equations by graphing you will be presented with or asked to graph 2 linear equations in the same coordinate plane. • When analyzing the graphs there are three possible scenarios as follows: • One Solution • No Solution • Infinite Solutions • Let’s see what they look like…

9. One Solution • When analyzing a graph with systems of equations, the lines will intersect. • The intersection point represents the solution of both equations • You can check the solution by substituting the ordered pair into both equations and you should get a true statement for both equations • Let’s look at an example…

10. y x One Solution • This is the graph of the equations y = -x y = x + 2 y = -x (-1, 1) • Notice that the lines intersect at the point (-1, 1) y = x + 2 • The ordered pair represents the solution to the system of equations

11. Checking the Solution • You can check to make sure that the ordered pair (-1, 1) is the solution by substituting the x and y-values into the original equations as follows: Equation #1 Equation #2 y = -x 1 = - (-1) 1 = 1 y = x + 2 1 = -1 + 2 1 = 1  

12. Observation • Observe that the slope of each equation is different. y = -x m = -1 y = x + 2 m = 1 • Without doing any work, we can analyze the slopes of the equations and predict how many solutions there will be to a system of equations • If the slopes are different, then there will be only one solution to the system of equations

13. No Solution • In a graph with 2 parallel lines there will be no solution to the system of equations • In this instance, the lines do not intersect. Therefore, no ordered pair will be the solution to both equations… • Let’s see what that looks like…

14. y x No Solution • This is a graph of the equations y = 2x + 4 y = 2x - 1 Notice that the lines are parallel and do not intersect Because the lines do not intersect there is no solution to this system of equations y = 2x + 4 y = 2x - 1

15. Observation • Observe the slope an y-intercepts of each equation. y = 2x + 4m = 2, b = 4 y = 2x - 1m = 2, b = -1 • Without doing any work, we can analyze the slopes and y-intercepts of the equations and predict how many solutions there will be to the system of equations • If the slopes are same, with different y-intercepts then there will be no solution to the system of equations

16. Infinite Solutions • In a system of equations where the graph of each line is the same you will have an infinite number of solutions. • Because the graphs of each line overlap, all points on each line will make the system of equations true, therefore you will have an infinite number of solutions • Let’s look at an example…

17. y x Infinite Solutions • This is a graph of the equations 2y = x + 6 y = ½ x + 3 2y = x + 6 y = ½ x + 3 Both equations have the same graph Any ordered pair on the lines will make both equations true

18. Observation • Observe that the slope an y-intercepts of each equation. 2y = x + 6m = ½ , b = 3 y = ½ x + 3m = ½, b = 3 • If you transform the first equation into the slope-intercept form you will see that both equations have the same slope and y-intercept • If the slopes and y-intercepts are the same then there will be an infinite number of solutions to the system of equations

19. Systems of Equations by Substitution • You can also solve a system of equations by the substitution method. • First, you need to know that you cannot solve an equation with two variables. • In the substitution method you substitute the value of one variable into the other equation… • Let’s see what that looks like…

20. Example Solve y = x – 5 y = 3 In this instance the second equations gives you the value of y. To solve, substitute the value of y into the first equation and solve algebraically as follows: y = x – 5 The solution to the system of equation is x = 8 and y = 3 The ordered pair that will make this system of equation true is (8, 3) 3 = x – 5 +5 + 5 8 = x

21. Your Turn • In the notes section of your notebook write and solve the system of equations using the substitution method. • y = 3x – 4 x = 0 • x + y = 8 y = 6 Solutions (0, -4) (2, 6)

22. Summary • In the notes section of your notebook summarize the key concepts covered in today’s lesson • Today we discussed: • Solving systems of equations by graphing – what are the solution scenarios? • Solving systems of equations by substituting – how does that work?

23. Assignment • Text p. 417 # 12 – 23 Reminder • This assignment is due tomorrow • I do not accept late assignments • You must show how you got your answers or no credit (no work = no credit)