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Solving Linear Systems

Solving Linear Systems. Trial and Error Substitution Linear Combinations (Algebra) Graphing. Linear System. Two or more equations Each is a straight line The solution = points shared by all equations of the system. Linear System. There may be one solution There may be no solution

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Solving Linear Systems

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

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

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

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

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

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

  7. Try ( 2,7) 6 (2) – (7) = 5 3 (2) + 7 = 13 Try ( 1,10) 6 (1) – 10 = 5 3 (1) + 10 = 13 Trial and Error 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. • + • + X • + (c) MathScience Innovation Center 2007

  8. Example: 6x – 4y = 10 3x + y = 2 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 (c) MathScience Innovation Center 2007

  9. 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 Substitution (c) MathScience Innovation Center 2007

  10. 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 Substitution (c) MathScience Innovation Center 2007

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

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

  13. Example: 6x – 4y = 10 3x + y = 2 If we draw a bar and add does any variable disappear? + 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 ! (c) MathScience Innovation Center 2007

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

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

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

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

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

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

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

  21. To use a graphing calculator, solve for y. Graph each equation: 6x – 4y = 10 3x + y = 2 Y1 = (10-6x)/(-4) Y2 = 2- 3x Graphing Simplifying is not necessary. (c) MathScience Innovation Center 2007

  22. Y1 = (10-6x)/(-4) Y2 = 2- 3x Graphing (c) MathScience Innovation Center 2007

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

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

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

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

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

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

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