Graphing Linear Equations and Inequalities: Proficiency Exam
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1 Connexions module: m Graphing Linear Equations and Inequalities: Proficiency Exam Denny Burzynski Wade Ellis This work is produced by The Connexions Project and licensed under the Creative Commons Attribution License Abstract This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. In this chapter the student is shown how graphs provide information that is not always evident from the equation alone. The chapter begins by establishing the relationship between the variables in an equation, the number of coordinate axes necessary to construct its graph, and the spatial dimension of both the coordinate system and the graph. Interpretation of graphs is also emphasized throughout the chapter, beginning with the plotting of points. The slope formula is fully developed, progressing from verbal phrases to mathematical expressions. The expressions are then formed into an equation by explicitly stating that a ratio is a comparison of two quantities of the same type (e.g., distance, weight, or money). This approach benets students who take future courses that use graphs to display information. The student is shown how to graph lines using the intercept method, the table method, and the slope-intercept method, as well as how to distinguish, by inspection, oblique and horizontal/vertical lines. This module contains the prociency exam for the chapter "Graphing Linear Equations and Inequalities in One and Two Variables". 1 Prociency Exam For the following problems, construct a coordinate system and graph the inequality. Exercise 1 (Solution on p. 7.) ( here 1 ) 6x + 4 > 14 Exercise 2 (Solution on p. 7.) ( here 2 ) 8 < x Exercise 3 (Solution on p. 7.) ( here 3 ) Plot the ordered pairs (3, 1), ( 2, 4), (0, 5), ( 2, 2). Version 1.4: Jun 1, :01 am GMT "Graphing Linear Equations and Inequalities: Graphing Linear Equations and Inequalities in One Variable" < 2 "Graphing Linear Equations and Inequalities: Graphing Linear Equations and Inequalities in One Variable" < 3 "Graphing Linear Equations and Inequalities: Graphing Linear Equations and Inequalities in One Variable" <
2 Connexions module: m Exercise 4 (Solution on p. 7.) ( here 4 ) As accurately as possible, label the coordinates of the points that have been plotted on the graph. Exercise 5 (Solution on p. 7.) ( here 5 ) What is the geometric structure of the graph of all the solutions to the equation 2y+3x = 4? Exercise 6 (Solution on p. 7.) ( here 6 ) In what form is the linear equation in two variables ax + by = c? Exercise 7 (Solution on p. 7.) ( here 7 ) In what form is the linear equation in two variables y = mx + b? Exercise 8 (Solution on p. 7.) ( here 8 ) If an ordered pair is a solution to a linear equation in two variables, where does it lie geometrically? 4 "Graphing Linear Equations and Inequalities: Plotting Points in the Plane" < 5 "Graphing Linear Equations and Inequalities: Graphing Linear Equations in Two Variables" 6 "Graphing Linear Equations and Inequalities: Graphing Linear Equations in Two Variables" 7 "Graphing Linear Equations and Inequalities: The Slope-Intercept Form of a Line" 8 "Graphing Linear Equations and Inequalities: Graphing Linear Equations in Two Variables"
3 Connexions module: m Exercise 9 (Solution on p. 7.) ( here 9 ) Consider the graph of y = 2 7x If we were to place our pencil at any point on the line and then move it horizontally 7 units to the right, how many units and in what direction would we have to move our pencil to get back on the line? For the following two problems, nd the slope, if it exists, of the line containing the following points. Exercise 10 (Solution on p. 7.) ( here 10 ) ( 6, 1) and (0, 8) Exercise 11 (Solution on p. 7.) ( here 11 ) ( 2, 8) and ( 2, 10) Exercise 12 (Solution on p. 7.) ( here 12 ) Determine the slope and y intercept of the line 3y + 2x + 1 = 0. Exercise 13 (Solution on p. 7.) ( here 13 ) As we look at a graph left to right, do lines with a positive slope rise or decline? For the following problems, nd the equation of the line using the information provided. Write the equation in slope-intercept form. Exercise 14 (Solution on p. 7.) ( here 14 )Slope = 4, y-intercept = 3. Exercise 15 (Solution on p. 7.) ( here 15 )Slope = 3 2, y-intercept = 4 3. Exercise 16 (Solution on p. 7.) ( here 16 )slope = 2, passes through ( 1, 2). 3 Exercise 17 (Solution on p. 7.) ( here 17 )slope = 7, passes through (0, 0). Exercise 18 (Solution on p. 8.) ( here 18 ) passes through the points (5, 2) and (2, 1). For the following problems, graph the equation of inequality. Exercise 19 (Solution on p. 8.) ( here 19 - here 20 ) y = 1 3 x 2 9 "Graphing Linear Equations and Inequalities: The Slope-Intercept Form of a Line" 10 "Graphing Linear Equations and Inequalities: The Slope-Intercept Form of a Line" 11 "Graphing Linear Equations and Inequalities: The Slope-Intercept Form of a Line" 12 "Graphing Linear Equations and Inequalities: The Slope-Intercept Form of a Line" 13 "Graphing Linear Equations and Inequalities: The Slope-Intercept Form of a Line" 14 "Graphing Linear Equations and Inequalities: Finding the Equation of a Line" < 15 "Graphing Linear Equations and Inequalities: Finding the Equation of a Line" < 16 "Graphing Linear Equations and Inequalities: Finding the Equation of a Line" < 17 "Graphing Linear Equations and Inequalities: Finding the Equation of a Line" < 18 "Graphing Linear Equations and Inequalities: Finding the Equation of a Line" < 19 "Graphing Linear Equations and Inequalities: Graphing Linear Equations in Two Variables" 20 "Graphing Linear Equations and Inequalities: Graphing Equations in Slope-Intercept Form" <
4 Connexions module: m Exercise 20 (Solution on p. 8.) ( here 21 - here 22 ) 5y 2x + 15 = 0 Exercise 21 (Solution on p. 8.) ( here 23 - here 24 ) 4 (x + y) = 8 21 "Graphing Linear Equations and Inequalities: Graphing Linear Equations in Two Variables" 22 "Graphing Linear Equations and Inequalities: Graphing Equations in Slope-Intercept Form" < 23 "Graphing Linear Equations and Inequalities: Graphing Linear Equations in Two Variables" 24 "Graphing Linear Equations and Inequalities: Graphing Equations in Slope-Intercept Form" <
5 Connexions module: m Exercise 22 (Solution on p. 8.) ( here 25 here 26 ) 3 2 y + 2 = 0 Exercise 23 (Solution on p. 9.) ( here 27 here 28 ) x = 2 Exercise 24 (Solution on p. 9.) ( here 29 ) 2x + 3y > 6 25 "Graphing Linear Equations and Inequalities: Graphing Linear Equations in Two Variables" 26 "Graphing Linear Equations and Inequalities: Graphing Equations in Slope-Intercept Form" < 27 "Graphing Linear Equations and Inequalities: Graphing Linear Equations in Two Variables" 28 "Graphing Linear Equations and Inequalities: Graphing Equations in Slope-Intercept Form" < 29 "Graphing Linear Equations and Inequalities: Summary of Key Concepts" <
6 Connexions module: m Exercise 25 (Solution on p. 9.) ( here 30 ) Reading only from the graph, determine the equation of the line. 30 "Graphing Linear Equations and Inequalities: Finding the Equation of a Line" <
7 Connexions module: m Solutions to Exercises in this Module Solution to Exercise (p. 1) x < 3 Solution to Exercise (p. 1) 14 < x 10 Solution to Exercise (p. 1) Solution to Exercise (p. 2) (0, 1), (3, 3), ( 3, 0), (2, 3) Solution to Exercise (p. 2) straight line Solution to Exercise (p. 2) general form Solution to Exercise (p. 2) slope-intercept Solution to Exercise (p. 2) It lies on the line. 2 units up 3 2 no slope; vertical line at x = 2 slope = 2 3, y-intercept is ( ) 0, 1 3 rise y = 4x 3 y = 3 2 x y = 2 3 x + 8 3
8 Connexions module: m y = 7x y = 1 3 x y = 1 3 x 2 Solution to Exercise (p. 4) Solution to Exercise (p. 4) 4 (x + y) = 8
9 Connexions module: m Solution to Exercise (p. 5) Solution to Exercise (p. 5) x = 2 Solution to Exercise (p. 5) Solution to Exercise (p. 6) y = 1 3 x + 3
10 Connexions module: m
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