1. (c) MathScience Innovation Center 2007 Solving Linear Systems Trial and Error Substitution Linear Combinations (Algebra) Graphing

2. (c) MathScience Innovation Center 2007 Linear System Two or more equations Each is a straight line The solution = points shared by all equations of the system

3. (c) MathScience Innovation Center 2007 Linear System There may be one solution There may be no solution There may be infinite solutions

4. (c) MathScience Innovation Center 2007 Linear System Consistent= there is a solution Inconsistent= there is no solution Independent= separate, distinct lines Dependent= same line

5. (c) MathScience Innovation Center 2007 Linear System Consistent, independent Inconsistent, independent Consistent, dependent

6. (c) MathScience Innovation Center 2007 Trial and Error Try any point and see if it satisfies every equation in the system (makes each equation true) Example: 6x – y = 5 3x + y = 13 Try ( 2,7) and try ( 1,10)

7. (c) MathScience Innovation Center 2007 Trial and Error Try ( 2,7) 6 (2) – (7) = 5 3 (2) + 7 = 13 Try ( 1,10) 6 (1) – 10 = 5 3 (1) + 10 = 13 + + + X Conclusion: Since (2,7) works and (1,10) does not work, (2,7) is a solution to the system and (1,10) is not a solution.

8. (c) MathScience Innovation Center 2007 Substitution Solve one equation for one variable and substitute into the other equations. Hint: Easiest to solve for a variable with a coefficient of 1 Example: 6x – 4y = 10 3x + y = 2

9. (c) MathScience Innovation Center 2007 Substitution Example: 6x – 4y = 10 3x + y = 2 Solve for y in bottom equation: 6x – 4y = 10 y = 2 – 3x Substitute for y in top equation: 6x – 4(2-3x) = 10 y = 2 – 3x

10. (c) MathScience Innovation Center 2007 Substitution Simplify top equation and solve for x: 6x – 4(2-3x) = 10 6x – 8 + 12 x = 10 18 x = 18 18x/18 = 18/18 Substitute for y in top equation: 6x – 4(2-3x) = 10 y = 2 – 3x

11. (c) MathScience Innovation Center 2007 Substitution So x = 1. Substitute for y in bottom equation: y = 2 – 3x y = 2 – 3(1) Y = -1 Final solution: ( 1, -1)

12. (c) MathScience Innovation Center 2007 Substitution Check your work: Final solution: ( 1, -1) Example: 6x – 4y = 10 3x + y = 2 Example: 6(1) – 4( -1) = 10 3(1) + -1 = 2 + +

13. (c) MathScience Innovation Center 2007 Linear Combinations (Algebra) Try adding the equations together so that at least one variable disappears Hint: You can multiply any equation by an integer to insure this happens ! Example: 6x – 4y = 10 3x + y = 2 + If we draw a bar and add does any variable disappear?

14. (c) MathScience Innovation Center 2007 Linear Combinations (Algebra) Example: 6x – 4y = 10 3x + y = 2 Multiply this equation by -2 or 4

15. (c) MathScience Innovation Center 2007 Linear Combinations (Algebra) Example: 6x – 4y = 10 3x + y = 2 Multiply this equation by -2 or 4 Multiplying by -2 yields 6x – 4y = 10 -6x + -2y = -4 + If we draw a bar and add does any variable disappear? Yes, x - 6 y = 6

16. (c) MathScience Innovation Center 2007 Linear Combinations (Algebra) Example: 6x – 4y = 10 3x + y = 2 Since - 6 y = 6, y = -1 Now, use substitution to find x 6x – 4 (-1) = 10 3x + (-1) = 2 X = 1

17. (c) MathScience Innovation Center 2007 Linear Combinations (Algebra) Multiplying by 4: 6x – 4y = 10 12x + 4y = 8 + If we draw a bar and add does any variable disappear? Yes, y 18 x = 18 Now, x = 1. Substitute x = 1 to find y. 6 (1) – 4y = 10 12 (1) + 4y = 8 So, y = -1

18. (c) MathScience Innovation Center 2007 Linear Combinations (Algebra) One last question 6x – 4y = 10 3x + y = 2 Is it easier to multiply this equation by -2 or 4 ? Most people are more successful when using positive numbers

19. (c) MathScience Innovation Center 2007 Graphing Graph each equation: 6x – 4y = 10 3x + y = 2 Note: this problem is difficult because the equations are not solved for y

20. (c) MathScience Innovation Center 2007 Graphing Graph each equation: 6x – 4y = 10 3x + y = 2 So it might be easiest to hand plot using the x and y intercepts.

21. (c) MathScience Innovation Center 2007 Graphing Graph each equation: 6x – 4y = 10 3x + y = 2 To use a graphing calculator, solve for y. Y 1 = (10-6x)/(-4) Y 2 = 2- 3x Simplifying is not necessary.

22. (c) MathScience Innovation Center 2007 Graphing Y 1 = (10-6x)/(-4) Y 2 = 2- 3x

23. (c) MathScience Innovation Center 2007 Which is the easiest method to solve this system? x = 4 2x + 3 y = 14 A. Substitution B. Linear Combinations (algebra) C. Graphing Why? One equation is already solved for x, ready for substitution.

24. (c) MathScience Innovation Center 2007 Which is the easiest method to solve this system? y = 2 x - 4 y = ¾ x + 5 A. Substitution B. Linear Combinations (algebra) C. Graphing Why? Both equations are already solved for y.

25. (c) MathScience Innovation Center 2007 Which is the easiest method to solve this system? 3 x – 2 y = 14 4x + 2 y = 21 A. Substitution B. Linear Combinations (algebra) C. Graphing Why? When you add them together, the y disappears.

26. (c) MathScience Innovation Center 2007 Which is the easiest method to solve this system? x – 9 y = 10 2x + 3 y = 7 A. Substitution B. Linear Combinations (algebra) C. Graphing Why? Substitution would not be difficult either, but graphing would be more difficult.

27. (c) MathScience Innovation Center 2007 If you use linear combinations, what would you multiply by and which equation would you use? x – 9 y = 10 2x + 3 y = 7 A. Top equation by -2 B. Bottom equation by 3 Which might be a wee tiny bit easier? B. Working with positive numbers may lead to fewer errors

28. (c) MathScience Innovation Center 2007 Match a system to the easiest solution method. Y = 2x + 1 Y = 1/3 x - 9 Substitution Linear Combinations (Algebra) Graphing A B C y = 2x + 1 4x – 19 y = 34 3 x – 5 y = 26 - 3 x + 4 y = 17