# Chapter 3 Linear Systems

## Chapter 3 Linear Systems

by Troy Cole
## 1. Graphing Systems of Equations

## 2. Chapter 3.1

## 3. System of equations: a set of two or more equations that use the same variables.

## 4. Linear System: a line that contains two identicle variables.

## 5. Independent system: has a unique solution.

## 6. Dependnt system: does not have a unique solution.

## 7. Ex. y= x + 3 y= -2x + 3

## 8. Inconsistant system: a system that does not have a solution.

## 9. Chapter 3.2

## 10. Solving systems Algebraically

## 11. Equivalent systems: systems that have the same solution.

## 12. Subsitution Ex. 4x + 3y = 4 2x - y = 7

## 13. Solve for "y" 2x - y = 7 y= 22 -7

## 14. Subsitute the "y" 4x + 3y =4 4x + 3(2x - 7) =4 4x + 6x - 21 =4 4x + 6x =25 x=2.5.

## 15. Solve for "y" again y= 2x - 7 y= 2(2.5) - 7 y= -2

## 16. Final Answer (2.5, -2)

## 17. Chapter 3.3

## 18. Systems of Inequalities

## 19. Ex. x - 2y < 6 y< -3/2x + 5

## 20. Chapter 3.4

## 21. Linear Programming

## 22. Linear Programming: identifies the minimum and maximum.

## 23. Objective Function: is how the minimum is modeled.

## 24. Constraints: are the limits on the variables.

## 25. Feasible Region: is the area on a graph that satisfies all the constraints.

## 26. Chapter 3.5

## 27. Graphs in Three Dimension

## 28. Coordinate Space: is adding a third axis, "z".

## 29. Ordered Triplets: (X,Y,Z)

## 30. Trace: when the graph of a pkane intersects one of the coordinate planes in a line.

## 31. Chapter 3.6

## 32. Systems with Three Variables

## 33. Three Systems x - 3y + 3z = -4 2x + 3y - z = 15 4x - 3y - z = 19

## 34. Paired for elimination x - 3y + 3z = -4 2x + 3y - z = 15 --------------------- 3x + 2z = 11

## 35. and....

## 36. Paired for elimination 2x + 3y - z = 15 4x - 3y - z = 19 ---------------------- 6x - 2z = 34

## 37. Solve for "x" 3x + 2z = 11 6x - 2z = 34 ----------------- 9x =45 x=5

## 38. Solve for "z" 3x + 2z =11 3(5) + 2z = 11 ------------------- 2z = -4 z=-2

## 39. Solve by subsitution x - 3y + 3z = -4 5 - 3y + 3(-2) = -4 5 - 3y - 6 = -4 -3y = -3 y=1

## 40. Final answer (5,1,-2)

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