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Int. Alg. Notes Test #3 Review Page 1 of 9 Section 4.1: Systems of Linear Equations in Two Variables A system of linear equations is a grouping of two or more linear equations, each of which contains one or more variables. A solution of a system of linear equations consists of values for the variables that are solutions to ALL of the equations in the system. Geometric/Visual Interpretation of a System of Two Linear Equations in Two Variables: INTERSECT: The lines intersect at one point, and thus the system has exactly one solution. This type of system is called consistent and the equations are called independent. PARALLEL: The lines never intersect (i.e., they are parallel to one another), and thus the system has no solutions. This type of system is called inconsistent. COINCIDENT: The lines lie on top of each other, and thus the system has infinitely many solutions. This type of system is called consistent and the equations are called dependent. Solving a System of Two Linear Equations by Graphing Graph both the lines. Read the coordinates of the intersection point off the graph. Check to see if those coordinates are the solution. Solving a System of Two Linear Equations Using Substitution Solve one of the equations for one of the unknowns. Substitute the expression for that unknown into the other equation. Solve the resulting equation in one unknown. Substitute that solution into the first equation to solve for the remaining variable. Check your answer. Solving a System of Two Linear Equations Using Elimination Multiply one or both equations by nonzero constants so that the coefficients of one of the variables are additive inverses. Add the two equations to obtain a new equation in one unknown. Solve the resulting equation in one unknown. Substitute that solution into either of the original equations and solve for the remaining variable. Check your answer. Algebra is: the study of how to perform multi-step arithmetic calculations more efficiently, and the study of how to find the correct number to put into a multi-step calculation to get a desired answer. Int. Alg. Notes Test #3 Review Page 2 of 9 Section 4.2: Problem Solving: Systems of Two Linear Equations Containing Two Unknowns Big Skill: You should be able to write out a model for a real-world situation involving two equations in two unknowns, then solve that system to get an answer. Section 4.3: Systems of Linear Equations in Three Variables Example of a system of three linear equations in three unknowns: y z 2 x x 2 y 3z 12 2 x 2 y z 9 A linear equation with three variables describes a two-dimensional plane embedded in three dimensions: When two planes intersect, the intersection is usually a line and when three planes intersect, the intersection is usually a point: y x Algebra is: the study of how to perform multi-step arithmetic calculations more efficiently, and the study of how to find the correct number to put into a multi-step calculation to get a desired answer. Int. Alg. Notes Test #3 Review Page 3 of 9 Geometric/Visual Interpretation of a System of Three Linear Equations in Three Variables: Exactly one solution: A consistent system with independent equations where the planes intersect at a single point. No solution: An inconsistent system where the planes are either all parallel, or intersect along parallel lines. Inconsistent systems yield a false equation (like 0 = 3) after trying to solve them. Infinitely many solutions: A consistent system with dependent equations where the planes all intersect along the same line, or are all coincident. Dependent systems yield one or more equations of 0 = 0 after applying Gaussian elimination. Example of a system of three linear equations in three unknowns that is in triangular form: x 2 y z 1 y 2z 5 z 3 Notice: the name triangular form comes from the “blank” triangular space in the lower left corner due to no x or y variables. The goal of solving a system of linear equations using elimination is to get the system into triangular form, because a triangular form system is really easy to solve using back-substitution. Key fact behind the technique of elimination: Multiplying an equation in a system by a constant, or adding two equations in a system together results in a new system of equations called a transformed system, and the solution to the transformed system is the same as the solution to the original system. Steps for Solving a System of Three Linear Equations in Three Unknowns Using Elimination Eliminate the x variable from the second and third equations using elimination. Eliminate the y variable from the third equation using elimination. Solve for z in the third equation. Substitute z into the second equation to find the solution for y, then substitute y and z into the first equation to find the solution for x. Rules for showing your work: Draw an arrow from one transformed system to the next, and write on the arrow what you did to transform the system. Any equation that is unchanged gets copied from one system to the next. Algebra is: the study of how to perform multi-step arithmetic calculations more efficiently, and the study of how to find the correct number to put into a multi-step calculation to get a desired answer. Int. Alg. Notes Test #3 Review Page 4 of 9 Example: y z 2 x x 2 y 3 z 12 2 x 2 y z 9 (1)(eqn. #1) is placed in row#1 2 x 2 y 2 z 4 y 4 z 14 4y z 5 (4)(eqn. #2) is placed in row#2 y z 2 x x 2 y 3 z 12 2 x 2 y z 9 (eqn. #1) (eqn. #2) is placed in row#2 2 x 2 y 2 z 4 4 y 16 z 56 4y z 5 (eqn. #2) (eqn. #3) is placed in row#3 y z 2 x y 4 z 14 2x 2 y z 9 2 x 2 y 2 z 4 4 y 16 z 56 17 z 51 (eqn. #3) ( 17) is placed in row#3 (2)(eqn. #1) is placed in row#1 2 x 2 y 2 z 4 y 4 z 14 2x 2 y z 9 (eqn. #1) (eqn. #3) is placed in row#3 2 x 2 y 2 z 4 y 4 z 14 4y z 5 2 x 2 y 2 z 4 4 y 16 z 56 z 3 Substitute into equation #2: 4 y 16 3 56 y 2 Substitute into equation #1: 2 x 2 2 2 3 4 x 1 Algebra is: the study of how to perform multi-step arithmetic calculations more efficiently, and the study of how to find the correct number to put into a multi-step calculation to get a desired answer. Int. Alg. Notes Test #3 Review Page 5 of 9 Section 4.4: Using Matrices to Solve Systems Example of a system of three equations in three unknowns (that happens to be in triangular form) and that system’s augmented matrix: 1 2 1 1 x 2 y z 1 y 2 z 5 0 1 2 5 0 0 1 3 z 3 Steps for Solving a System of Three Linear Equations in Three Unknowns Using an Augmented Matrix Eliminate the x coefficients from the second and third rows using elimination. Eliminate the y coefficient from the third row using elimination. Make the coefficient of z in the third equation equal to one by dividing by the appropriate number. Convert the triangular matrix back into a system of equations. Substitute z into the second equation to find the solution for y, then substitute y and z into the first equation to find the solution for x. Rules for showing your work: Draw an arrow from one transformed matrix to the next, and write on the arrow what you did to transform the matrix. Any row that is unchanged gets copied from one system to the next. Algebra is: the study of how to perform multi-step arithmetic calculations more efficiently, and the study of how to find the correct number to put into a multi-step calculation to get a desired answer. Int. Alg. Notes Test #3 Review Page 6 of 9 Example: y z 2 x x 2 y 3 z 12 2 x 2 y z 9 convert to augmented matrix 2 2 2 4 0 1 4 14 0 4 1 5 (4)(row #2) is placed in row#2 1 1 1 2 1 2 3 12 0 2 1 9 (1)(row #1) is placed in row#1 2 2 2 4 0 4 16 56 0 4 1 5 (row #2) (row #3) is placed in row#3 1 1 1 2 1 2 3 12 2 2 1 9 (row #1) (row #2) is placed in row#2 2 2 2 4 0 4 16 56 0 0 17 51 (row #3) (17) is placed in row#3 1 1 1 2 0 1 4 14 2 2 1 9 (2)(row #1) is placed in row#1 2 2 2 4 0 4 16 56 0 0 1 3 convert back to a system of equations 2 2 2 4 0 1 4 14 2 2 1 9 (row #1) (row #3) is placed in row#3 2 x 2 y 2 z 4 4 y 16 z 56 z 3 2 2 2 0 1 4 2 4 1 4 14 5 Substitute into equation #2: 4 y 16 3 56 y 2 Substitute into equation #1: 2 x 2 2 2 3 4 x 1 Algebra is: the study of how to perform multi-step arithmetic calculations more efficiently, and the study of how to find the correct number to put into a multi-step calculation to get a desired answer. Int. Alg. Notes Test #3 Review Page 7 of 9 Section 4.5: Determinants and Cramer's Rule Definition: Determinant of a 2 2 matrix a b a b Suppose that a, b, c, and d are real numbers. The determinant of the 2 2 matrix , written as , c d c d a b is ad bc . c d Cramer’s Rule for solving a system of two linear equations in two unknowns ax by s The solution to the system of equations is given by cx dy t s x b a s Dy t d c t D x and y a b a b D D c d c d Provided that D a b c d ad bc 0 Algebra is: the study of how to perform multi-step arithmetic calculations more efficiently, and the study of how to find the correct number to put into a multi-step calculation to get a desired answer. Int. Alg. Notes Test #3 Review Definition: Determinant of a 3 3 matrix a1,1 a1,2 The determinant of the 3 3 matrix a2,1 a2,2 a3,1 a3,2 Page 8 of 9 a1,3 a1,1 a2,3 , written as a2,1 a3,3 a3,1 a1,2 a1,3 a2,2 a2,3 , is calculated using the a3,2 a3,3 determinants of 2 2 matrices as follows: a1,1 a1,2 a1,3 a2,1 a2,2 a2,3 a1,1 a3,1 a3,2 a3,3 a2,2 a2,3 a3,2 a3,3 a1,2 a2,1 a2,3 a3,1 a3,3 a1,3 a2,1 a2,2 a3,1 a3,2 . Notice a pattern: the 2 2 determinants are what remains after the row and column corresponding to the matrix element in front of the 2 2 determinant is blocked out. Cramer’s Rule for solving a system of three linear equations in three unknowns a1 x b1 y c1 z d1 The solution to the system of equations a2 x b2 y c2 z d 2 a x b y c z d 3 3 3 3 with a1 b1 c1 d1 b1 c1 a1 d1 c1 a1 b1 d1 D a2 b2 c2 0 , Dx d 2 b2 c2 , Dy a2 d 2 c2 , and Dz a2 b2 d 2 a3 b3 c3 d3 b3 c3 a3 d3 c3 a3 b3 d3 is given by D D D x x , y y , and z z . D D D Algebra is: the study of how to perform multi-step arithmetic calculations more efficiently, and the study of how to find the correct number to put into a multi-step calculation to get a desired answer. Int. Alg. Notes Test #3 Review Page 9 of 9 Section 4.6: Systems of Linear Inequalities Solving a System of Linear Inequalities: Graph each inequality separately Choose the region that is the intersection of all the inequalities. Example: 2x The graph of the solution of the system 3x 5 x 4y 8 2y 4 is: 3y 6 Algebra is: the study of how to perform multi-step arithmetic calculations more efficiently, and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.