1. Splash Screen

2. Example 2 Solve by Graphing Solve the system of equations by graphing. x – 2y = 0 x + y = 6 The graphs appear to intersect at (4, 2). Write each equation in slope-intercept form.

3. Example 2 Solve by Graphing Check Substitute the coordinates into each equation. x – 2y= 0x + y=6Original equations 4 – 2(2)= 04 + 2=6Replace x with 4 and y with 2. ?? 0=06=6Simplify. Answer: The solution of the system is (4, 2).

4. Example 2 Which graph shows the solution to the system of equations below? x + 3y=7 x – y=3 A.C. B.D.

5. Concept

6. Example 3 Classify Systems A. Graph the system of equations and describe it as consistent and independent, consistent and dependent, or inconsistent. x – y = 5 x + 2y = –4 Write each equation in slope-intercept form.

7. Example 3 Classify Systems Answer: The graphs of the equations intersect at (2, –3). Since there is one solution to this system, this system is consistent and independent.

8. Example 3 Classify Systems B. Graph the system of equations and describe it as consistent and independent, consistent and dependent, or inconsistent. 9x – 6y = –6 6x – 4y = –4 Write each equation in slope-intercept form. Since the equations are equivalent, their graphs are the same line.

9. Example 3 Classify Systems Answer: Any ordered pair representing a point on that line will satisfy both equations. So, there are infinitely many solutions. This system is consistent and dependent.

10. Example 3 Classify Systems C. Graph the system of equations and describe it as consistent and independent, consistent and dependent, or inconsistent. 15x – 6y = 0 5x – 2y = 10 Write each equation in slope-intercept form.

11. Example 3 Classify Systems Answer : The lines do not intersect. Their graphs are parallel lines. So, there are no solutions that satisfy both equations. This system is inconsistent.

12. Example 3 Classify Systems D. Graph the system of equations and describe it as consistent and independent, consistent and dependent, or inconsistent. f(x) = –0.5x + 2 g(x) = –0.5x + 2 h(x) = 0.5x + 2

13. Example 3 Classify Systems Answer: f(x) and g(x) are consistent and dependent. f(x) and h(x) are consistent and independent. g(x) and h(x) are consistent and independent.

14. Example 3 A. Graph the system of equations below. What type of system of equations is shown? x + y = 5 2x = y – 5 A.consistent and independent B.consistent and dependent C.consistent D.none of the above

15. Example 3 B. Graph the system of equations below. What type of system of equations is shown? x + y = 3 2x = –2y + 6 A.consistent and independent B.consistent and dependent C.inconsistent D.none of the above

16. Example 3 C. Graph the system of equations below. What type of system of equations is shown? y = 3x + 2 –6x + 2y = 10 A.consistent and independent B.consistent and dependent C.inconsistent D.none of the above

17. Example 3 A.f(x) and g(x) are consistent and dependent. B.f(x) and g(x) are inconsistent. C.f(x) and h(x) are consistent and independent. D.g(x) and h(x) are consistent. D. Graph the system of equations below. Which statement is not true? f(x) = x + 2 g(x) = x + 4

18. Concept

19. Example 5 Solve by Using Elimination Use the elimination method to solve the system of equations. x + 2y = 10 x + y = 6 In each equation, the coefficient of x is 1. If one equation is subtracted from the other, the variable x will be eliminated. x + 2y=10 (–)x + y= 6 y= 4Subtract the equations.

20. Example 5 Solve by Using Elimination Now find x by substituting 4 for y in either original equation. x + y=6Second equation x + 4=6Replace y with 4. x= 2Subtract 4 from each side. Answer:The solution is (2, 4).

21. Example 5 A.(2, –1) B.(17, –4) C.(2, 1) D.no solution Use the elimination method to solve the system of equations. What is the solution to the system? x + 3y = 5 x + 5y = –3

22. Example 6 Read the Test Item You are given a system of two linear equations and are asked to find the solution. No Solution and Infinite Solutions Solve the system of equations. 2x + 3y = 12 5x – 2y = 11 A. (2, 3) B. (6, 0) C. (0, 5.5) D. (3, 2)

23. Example 6 No Solution and Infinite Solutions x =3x =3 Solve the Test Item Multiply the first equation by 2 and the second equation by 3. Then add the equations to eliminate the y variable. 2x + 3y=124x + 6y=24 Multiply by 2. Multiply by 3. 5x – 2y=11(+)15x – 6y=33 19x =57

24. Example 6 Replace x with 3 and solve for y. No Solution and Infinite Solutions 2x + 3y=12First equation 2(3) + 3y=12Replace x with 3. 6 + 3y=12Multiply. 3y=6Subtract 6 from each side. y=2Divide each side by 3. Answer:The solution is (3, 2). The correct answer is D.

25. Example 6 Solve the system of equations. x + 3y = 7 2x + 5y = 10 A. B.(1, 2) C.(–5, 4) D.no solution

26. Homework P. 141 # 3 – 11 odd, 19 – 25 odd, 31 – 41 odd, 51 – 57 odd

27. End of the Lesson