1. LINEAR SYSTEMS Chapter 3

2. Definitions System  System (form Latin systema)- “set of interacting or interdependent entities forming an integrated whole”  In other words, interacting parts that make up the whole Examples of Systems:  Solar System  Circulatory System  Operating System

3. System of Linear Equations System of Linear Equations- set of two or more equations Examples:  3x – y = 1 and 2x + y = 4  x + y = 5 and 3x – 2y = 20 Possible Solutions  No solution (parallel lines)  Infinite ordered pairs (coinciding lines)  One solution (lines intersect at one point)

4. Solving System of Linear Equations Graphically Graph both linear equations in the same coordinate plane  Solve each linear equation for y (slope-intercept form)  Plot using the slope and y-intercept  Use a straight edge to carefully draw each line Determine the intersection, if any The solution is the set of values for the variables that makes all the equations true For 2 equations with 2 variables, the solution is the set of ordered pairs that represent the intersection of the two graphs

5. More Definitions Inconsistent System of Equations- has no solution {empty set}; examples: parallel or skew lines Consistent System of Equations- has at least one solution Independent System of Equations- has exactly one solution Dependent System of Equations- has an infinite number of solutions

6. Summary Graphs of Equations Slopes of LinesName of System of Equations Number of Solutions Lines intersectDifferent slopesConsistent and Independent One Lines coincideSame slopes, same intercepts Consistent and Dependent Infinite Lines parallelSame slope, different intercepts InconsistentNone

7. Consistent and Dependent Systems Example:  y = -3x + 5 and 9x + 3y = 15  Both equations in slope-intercept form are y = -3x +5  Both lines have the same slope and intercepts  Therefore, the system is consistent and dependent.

8. Solve System of Equations by Substitution Examples Substitution- solve for one variable in one equation and substitute this into the second equation. Line 1: y = 2x + 1 Line 2: 2y = 3x – 2 Find the solution to this system of linear equations. x = -4 y = -7 Solution is (-4, -7)

9. Solving Systems if Linear Equations by Elimination Want to add two equations together so that one term cancels out Multiply one equation by the value that will cause one term to drop out when the equations are added together Example: 2x + y = 9 + 3x – y = 16 5x + 0 + 25 x = 5, y = -1; solution (5, -1)

10. What if we don’t have y and –y or x and –x to cancel out? Example: 2x + 4y = 8 x + 3y = 6 What can we multiply the second equation by in order to eliminate the x’s? We need the second equation to have a -2x term. Multiply the second equation by -2. -2x -6y = -12 Then add the two equations together.

11. Elimination Example (continued) 2x + 4y = 8 -2x -6y + -12 -2y = -4 y = 2 x = 0 Verify solution by plugging values of x and y into the original equations.

12. Homework p.139 #17, 19, 21, 23, 25, 28 Systems of Two Equations worksheet #8, 11-14, 18, 21-24