2. Review By: Kevin Alvernaz Henze Gustave Elias Peralta Brielle Wanamaker

3. Section 2.1 Solving Systems of Equations Vocabulary System of Equations - A set of two or more equations Solution – An ordered pair representing the solution common to both equations in a system Consistent System – Has at least 1 solution Independent system – Has exactly 1 solution Inconsistent system – There is no solution

4. Solve by Graphing Equations must be in y- intercept form ( y = mx + b) where m is the slope and b is the y – intercept The Y variable cannot be negative For Example 3x – 2y = -6 becomes y = 3/2x + 3 x + y = -2 becomes y = -x -2 When the graphs... Intersect, there is one solution Are on the same line, there are many solutions Are parallel, there are no solutions *graphs with the same slope and different intercepts will always be parallel*

5. Solve by Elimination In elimination one variable is eliminated so that the equation can be solved in terms of the other 5(1.5x + 2y = 20) BECOMES7.5x + 10y = 100 The 10y & - 10y will cancel 2(2.5x – 5y = -25)5x – 10y = - 50 out leaving only the x variable This leaves 12.5x = 50 which means x = 4 Once an answer is reached for one variable you can plug it into any of the original equations to find the second variable 1.5x + 2y = 20 ====> 1.5(4) + 2y = 20 ====> 2y = 14 =====> y = 7

6. Solve by Substitution In substitution one of the equations is set equal to a variable and substituted into the second equation For Example 2x + 3y = 8 x- y = 2 becomes x = y + 2 After Substitution 2(y + 2) + 3y = 8 ==> 5y = 4 ==> y = 4/5 Once a solution is reached it can be plugged into any of the other original equations x – y = 2 ===> x – 4/5 = 2 ===> x = 14/5

7. Homework PG 71 #’s 21-24 21. 5x –y = 1622. 3x -5y = -823. y = 6 – x 2x+3y = 3 x + 2y = 1 x = 4.5 + y 24. 2x +3y = 3 12x -15y = -4

8. Section 2.2 Solving Systems of Equations with three variables

9. Solve By Elimination Elimination with three variables works the same as with two, your goal is to eliminate one variable at a time. For Example x – 2y + z = 15Choose one variable and equation 2x + 3y – 3z = 1 to use for the remaining two equations. For this example 4x + 10y – 5z = -3I have chosen the equation in bold.

10. -2 (x – 2y + z = 15) and -4(x – 2y + z = 15) 2x +3y -3z = 1 4x + 10y – 5z = -3 -2x + 4y – 2z = -30(2x’s cancel out) -4x + 8y – 4z = -60 (4x’s cancel out) 2x + 3y -3z = 1 4x +10y – 5z = -3 7y – 5z = -29 *Now use the elimination method with these two equations* 18y + 9z = -63 -9(7y – 5z = -29) ===> -63y +45z = 261(45z’s cancel out) 5(18y – 9z = -63) ===> 90y – 45z = -315 Leaving Only ===> 27y = -54 Therefore y = -2 Substitute Y into one of the previous equations 7(-2) – 5z = -29 Z = 3 Then go back to one of the original equations and plug in the value of Z and Y to find X X – 2(-2) +3 = 15 X = 8Solution x = 8, y = -2, z = 3

11. Solve by Substitution The same problem can be solved by using the substitution method x – 2y + z = 15 becomes x = 2y – z + 15 now substitute it for both remaining equations 2x + 3y – 3z = 1 4x + 10y – 5z = -3 2(2y – z + 15) +3y – 3z = 1and 4(2y – z +15)+10y – 5z = -3 4y – 2z +30 +3y -3z = 18y – 4z +60 +10y - 5z = -3 7y – 5z = -29Now use substitution with these two equations 18y – 9z = -63 18y – 9z = -63 ===> z = 2y + 7 7y – 5z = -29 7y -5(2y+7) = -29 7y – 10y – 35 = -29 y = -2Now substitute into the previous equation 7(-2) – 5z = -29 ==> z = 3 Finally substitute the values for z and y in any original equation X – 2(-2)+3 = 15 ==> x = 8 Solution x = 8, y = -2, z = 3

12. Homework Pg 74 4-6 Solve each system of equations using the substitution method or elimination method 4. 4x +2y +z = 75. x – y – z = 7 6. 2x – 2y +3z = 6 2x +2y – 4z = -4 –x +2y – 3z = -12 2x – 3y +7z = -1 X +3y – 2z = -8 3x – 2y +7z = 30 4x – 3y +2z = 0

13. Chapter 2 Section 3 Modeling Real-World Data with Matrices

14. Vocabulary 1.Matrix- rectangular array of terms. 2. Element- the terms in the matrices. 3. M x N Matrix- a matrix with “m” rows and “n” columns. 4. Dimensions- are the “m” and “n”, or the rows and columns. 5. Row Matrix- a matrix that has only one single row. 6. Column Matrix- a matrix with only one single column. 7. Square Matrix- has the same number of rows as columns. 8. Nth order- is when “n” is the number of rows and columns in a matrix. 9. Zero Matrix- is

15. 2.5 Determinants and Multiplicative Inverses of Matrices Vocabulary Determinant- of Minor- an element of any nth-order determinant is a determinant of order (n-1). Identity matrix for multiplication- for any square matrix A is the matrix I, such as that AI=A and IA=A. Represented as Inverse matrix- can be designated as

16. Homework Page 102-103 #’s 14, 18, 23, 27 & 34

17. 2.6 Solving Systems of Linear Inequalities Find the maximum and minimum values of f(x, y) x – y= 2 for the polygonal convex set determined by the system of inequalities.

18. Homework Page 110 9-15

19. The End