MATH 115: Review for Chapter 6

Transcription

1 MATH 115: Review for Chapter 6 In order to prepare for our test on Chapter 6, ou need to understand and be able to work problems involving the following topics: I SYSTEMS OF LINEAR EQUATIONS CONTAINING VARIABLES. A. Can ou solve a sstem of equations graphicall? B. Can ou solve a sstem of equations b substitution? C. Can ou solve a sstem of equations b elimination? D. Do ou know what a sstem of equations is called if no solution exists? (Inconsistent sstem) Can ou explain the graphical interpretation of this? (Parallel Lines, do not intersect) Can ou identif that a sstem is inconsistent when attempting to solve it algebraicall? E. Do ou know what the equations are called if there are infinitel man solutions? (Dependent) Can ou explain the graphical significance of this? (Each equation has the same graph) Can ou solve a sstem of equations algebraicall when the equations are dependent? Solve the following sstems of equations graphicall. If there is one solution point, state its coordinates to decimal places. If the sstem has no solution, state that the sstem is inconsistent. If the sstem has infinitel man solutions, state that the equations are dependent and describe the solution set. (1) x 5 4x 5 () x x () x 5x 1 (4) x 6 9x 4 49 (5) x 5 4x 6 10 Solve the following sstems of equations b substitution. If there is one solution point, state its coordinates as exact integers, fractions, or mixed numbers. If the sstem has no solution, state that the sstem is inconsistent. If the sstem has infinitel man solutions, state that the equations are dependent and describe the solution set. (6) x 5 4x 5 (7) x 4x 6 (8) c d 5c d 1 (9) x 5 x (10) x x 5 7 Revised Januar 016 1

2 Solve the following sstems of equations b elimination. If there is one solution point, state its coordinates as exact integers, fractions, or mixed numbers. If the sstem has no solution, state that the sstem is inconsistent. If the sstem has infinitel man solutions, state that the equations are dependent and describe the solution set. (11) x 5 4x 5 (1) a b 1 5a 4b 7 (1) x 5x 1 (14) 7 1 x 8 1 x (15) 6x x 6 10 II SYSTEMS OF LINEAR EQUATIONS WITH VARIABLES (Algebraic solution) A. Do ou know the rules for obtaining an equivalent sstem of equations? 1. Interchange an equations of the sstem.. Multipl (or divide) each side of an equation b the same nonzero constant.. Replace an equation in the sstem b the sum (or difference) of that equation and a nonzero multiple of an other equation in the sstem. B. Do ou know how to solve a sstem of linear equations containing variables algebraicall, using the rules for obtaining an equivalent sstem? C. Can ou determine that a sstem is inconsistent, and state that there is no solution? D. Can ou determine that the equations are dependent, and describe the infinite number of solutions in the solution set? Solve the following sstems of equations algebraicall. If there is one solution point, state its coordinates as exact integers, fractions, or mixed numbers. If the sstem has no solution, state that the sstem is inconsistent. If the sstem has infinitel man solutions, state that the equations are dependent and describe the solution set. (16) x z 1 z x z 0 (17) x z 5 x 4 z 0 x z (18) x z 1 x z x z 5 (19) z 5 x 4 z x 6 z 8 (0) x z 0 x 6 z 1 x z (1) x z 0 x z 0 x 4 5z 0 () x x x 1 () x z 1 x 5 4z 4 x z (4) x z 8 x 4 z 10 x z Revised Januar 016

3 IV APPLICATIONS OF SYSTEMS OF LINEAR EQUATIONS A. Can ou identif the variables and relationships that are the ke elements of a word problem? B. Can ou construct a sstem of equations containing each of the ke variables? C. Can ou solve the sstem of equations, check the solution, and write a clear and complete statement that describes the ke elements of the solution? Solve the following applications using sstems of equations: (5) Determine the equation of the parabola of the form passes through the points,1 and 1,. x bx c that (6) Find a quadratic function g x such that g 0, g 5. g 1, and (7) A furniture factor manufactures custom-designed tables in two grades. Grade A tables require 1 hour of assembling and hours finishing. Grade B tables require 1 hour of assembling and 1 hour finishing. If there are 800 hours available for assembl and 100 hours available for finishing, how man of each grade of table should be manufactured? (8) A furniture factor manufactures custom-designed tables in grades. Grade A tables require 1 hour of assembling, 1.5 hours of surface preparation, and hours of finishing. Grade B tables require 1 hour of assembling, 1 hour of surface preparation, and one-half hour of finishing. Grade C tables require 1 hour of assembling and 1 hour of surface preparation, without an finishing. If 18 hours are available for assembl, for surface preparation, and 18 hours for finishing, how man of each grade of table should be manufactured? VII SYSTEMS OF LINEAR INEQUALITIES A. Can ou graph a linear inequalit in variables, b hand and using a graphing calculator? B. Can ou graph a sstem of linear inequalities in variables, b hand and using a calculator? Sketch the graphs of the following linear inequalities: (9) x 6 (0) x 0 Revised Januar 016

4 Sketch the graphs of the following sstems of linear inequalities: (1) x 5 () x x 0 () x x 6 (4) x 0, 0 x 4 1 x Revised Januar 016 4

5 Answers: (1) () () (4) (5) (6) 1, 1 (7) inconsistent sstem, no solution (8) 1,1 4 4 (9) 5 1, 7 7 (10) 5, (11) 1, 1 (1) 4 5, 7 7 (1) 1,1 4 4 (14) 1 1,1 8 (15) dependent,, 5 x x (16) 5,, (17) 1,, Revised Januar 016 5

6 (18) inconsistent sstem, no solution (19) dependent,,, x x z z z an real number (0) 5 1,, (1) dependent, x z x,, an real number z 0 () inconsistent sstem, no solution () dependent, x z x,, z 1 z z an real number (4) 1 7, 6,0 (5) 5 17 x x 4 4 (6) g x x 4 (7) 400 tables of each grade (8) 8 Grade A, 4 Grade B, and 6 Grade C tables (9) x (0) x Revised Januar 016 6

7 (1) () () (4) Revised Januar 016 7