1. Chapter 3 Linear Equations and Functions TSWBAT find solutions of two variable open sentences, and graph linear equations and points in two variables.

2. Two Variable Equations  Open Sentence – a statement or equation that can have more than one solution.  Open Sentence in Two Variables – A statement that can have more than one solution but contains two different variables.

3. Two Variable Equations  Solution Set – Set of all solutions to a given problem.  Solution – To a two variable sentence is an Ordered Pair.  Ordered Pair – Solution to a two variable sentence or the value of the x and y terms to a point on a graph. An ordered pair is written with the x-term first and the y-term second. EX (X, Y); (2,5); (-5,3); (8,-1)

4. Graphing  xy- Coordinate Plane – The normal two directional plane on which an open sentence in two variables can be graphed.

5. Graphing  Plane Rectangular Coordinate System or Cartesian coordinate system– Made up of two number lines that intersect at right angles at the point O. This is also named the Cartesian system after French mathematician Rene Descartes who introduced the idea of coordinates.

6. Graphing  Origin – The intersection point of the two number lines in the coordinate system labeled as point O.  x-axis – The horizontal number line in a Coordinate System.  y-axis – The vertical number line in a Coordinate System.

7. Graphing  Quadrants – The four sections made up by the intersection of the x and y axis in the coordinate system.  Plotting – In graphing points and lines on the coordinate system we call graphing a point plotting the point or placing the point on the graph.  Domain – The values for which x can be in a two variable sentence or on the coordinate system.  Range – The values for which y can be in a two variable sentence or on the coordinate system.

8. Graphing  One – to – One Correspondence – between ordered pairs and points on the plane can be summarized as:  1. There is exactly one point on the plane associated with each ordered pair.  2. There is exactly one ordered pair associated with each point on the plane.

9. Graphing  Graph - of an open sentence in two variables is the set of all points in the coordinate plane that satisfies the sentence.  Linear Theorem - The graph of every equation of the form : Ax + By = C, when A and B are not both 0, is a line. Similarly every line in the coordinate plane is the graph of an equation in this form.

10. Linear vs. Non-Linear  Linear – Forms a line  Examples of linear: 5x + 3y = -8,  Non-Linear – Does not form a line.  Examples of Non-linear: 2x +3y 2 = 4, xy = 2,

11. X and Y Intercepts  To solve for x-Intercept 1. Solve equation for X. 2. Substitute 0 in for y. 3. Solve  To solve for y-Intercept 1. Solve equation for y. 2. Substitute 0 in for x. 3. Solve

12. Graph a Line  It is best to have 3 points on the line, but you only need 2.  The easiest way is to graph the two intercepts and then plot the third point you are given or find to determine the direction of the line.

13. Examples  Finding Solutions to two variable equations

14. Graphing  Graphing Points and Lines

15. Graphing  Finding X and Y Intercepts

16. Chapter 3 TSWBAT Find Slope of a line, and graph a line given the slope and point on the line.

17. Slope  Slope of a Line L = where  Horizontal Line – Slope = 0  Vertical Line – No Slope  Coefficient – number or numerical factor in front of a variable. In the y-equals equation the coefficient in front of the x- term is the slope of the line.

18. Slope Theorems  Theorem 2 – The slope of the line Ax + By = C where is. Ax + By = C where is.  Theorem 3 – Let P(x 1,y 1 ) be a point and m a real number. There is one and only one line L through P having the slope m. An equation of L is y – y 1 = m(x – x 1 ).

19. Slope Generalizations  The slope of a line rises if m is positive.  The slope of a line falls if m is negative.  The larger is, the steeper the line is.

20. Slope  Examples – Finding Slope

21. Chapter 3 TSWBAT find an equation of a line given the slope and a point on the line, given two points, or given the slope and y- intercept.

22. Equations for a Line  Standard Form of the equation of a line is Ax + By = C with A, B, and C being integers.  There are two other forms for the equation of a line however.  Point-Slope form – the equation is then y – y1 = m(x – x1).  Slope-Intercept form – the equation is y = mx + b.

23. Finding Equation of a Line  Examples – Standard Form

24. Finding Equation of a Line  Examples – Point Slope Form

25. Finding Equation of a Line  Examples – Slope Intercept Form

26. Chapter 3 TSWBAT find equations of parallel and perpendicular lines, find linear functions and graph them and determine if relations are functions.

27. Parallel Lines  Parallel Lines have the same slope and never intersect.  Example

28. Perpendicular Lines  Have the opposite reciprocal slope of the other line.  These lines meet at only one point in a 90 degree angle.  Example

29. Functions and Relations  Function – a correspondence between two sets, D and R, that assigns to each member of D exactly one member of R. (One to One Correspondence).  Example:  Domain of the Function – is the Set D.  Example:  Range of the Function – is the Set R.  Example:

30. Functions and Relations  Values of a function – the members of the range assigned to a member of the domain.  Example: the function f assigns 2 the value 4.  Functional notation – f(x)=C Example: f(2)=4  Linear Functions – A function f that can be defined by the equation f(x)=mx+b where x, m, and b are real numbers, and the graph of f is the graph of the line y=mx+b with slope m and y-intercept b.  Example:

31. Functions and Relations  Constant Function – A function where m=0 and is thus f(x)=b for all x.  Example:  Is this a Horizontal or Vertical Line?  Rate of change m = slope of a line =

32. Functions and Relations  Relation – Any set of ordered pairs. A function is a relation but not all relations are functions. A relation can contain two or more ordered pairs with the same x and/or y values. A function can contain two or more ordered pairs with the same y value only.  Example:

33. Functions and Relations  Vertical - Line Test – a test to determine if a given relation is a function. This test says a relation is a function if and only if a vertical line intersects the graph of the relation at most one time.  Example:

34. Chapter 3 TSWBAT Solve systems of Linear Equations by 1. Linear Combinations, 2. Substitution, 3 Graphing.

35. Systems of Linear Equations  A system of linear equations or linear system – a set of linear equations in the same two variables. Example

36. Solutions to a Systems of Linear Equations  1. simultaneous solution - an ordered pair that satisfies both equations at their point of intersection.  Example

37. Solutions to a Systems of Linear Equations  2. the null set for two lines that are parallel.  Example

38. Solutions to a Systems of Linear Equations  3. a line if the set of linear systems is a group of coinciding lines.  Example

39. Systems of Linear Equations  Equivalent systems – systems of linear equations that have the same solution set.  Example  Linear Combination – the addition of two equations.  Example

40. Systems of Linear Equations  Consistent system – a system that has at least one solution.  Example  Inconsistent system – a system with no solution and lines that are inconsistent.  Example

41. Systems of Linear Equations  Dependent system – a system that has an infinite number of solutions and the lines are coinciding.  Example

42. Transformations  1.Replacing an equation by an equivalent expression. – That is multiplying each side of an equation by the same non-zero number.

43. Transformations  2. Substituting for one variable in an equation for that variable obtained from another equation in the system.

44. Transformations  3. Replacing any equation by the sum of that equation and another equation in the system. – That is add left sides, right sides, and then equate the results.

45. Systems of Linear Equations  Three methods to solve  1. Linear Combination  2. Substitution  3. Graphing

46. Systems of Linear Equations  Example Linear Combination

47. Systems of Linear Equations  Example Substitution

48. Systems of Linear Equations  Example Graphing

49. Chapter 3 TSWBAT solve linear inequalities and systems of linear inequalities.

50. Linear Inequalities  Linear Inequality in Two Variables – is when the equals sign in a linear equation in two variables is replaced by an inequality symbol like, or  Boundary – The linear equation from which the inequality was formed.  Solution – a shaded region defined by the inequality symbol and boundary.

51. Linear Inequalities  Open Half-Plane – When the boundary line is not included in the solution and is shown as a dashed line (when we have ).  Closed Half-Plane – When the boundary line is included in the solution and is shown as a solid line (when we have or ).

52. Linear Inequalities  Example -

53. System of Linear Inequalities  System of Inequalities – Two or more linear inequalities working together as a set.  Solution to a system of Inequalities – is the region where ALL inequalities have a shaded region as a solution.

54. System of Linear Inequalities  Example