Review By: Kevin Alvernaz Henze Gustave Elias Peralta Brielle Wanamaker

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

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*

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

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

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

Section 2.2 Solving Systems of Equations with three variables

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.

-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 = (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

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 y -3z = 18y – 4z y - 5z = -3 7y – 5z = -29Now use substitution with these two equations 18y – 9z = y – 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

Homework Pg Solve each system of equations using the substitution method or elimination method 4. 4x +2y +z = 75. x – y – z = x – 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

Chapter 2 Section 3 Modeling Real-World Data with Matrices

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

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

Homework Page #’s 14, 18, 23, 27 & 34

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.

Homework Page

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