ca_m3 - MAC 1105 Module 3 System of Equations and...

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Unformatted text preview: MAC 1105 Module 3 System of Equations and Inequalities Rev.S08 Learning Objectives Upon completing this module, you should be able to: 1. 2. 3. 4. 5. 6. Evaluate functions of two variables. Apply the method of substitution. Apply the elimination method. Solve system of equations symbolically. Apply graphical and numerical methods to system of equations. Recognize different types of linear systems. Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 2 Learning Objectives (Cont.) 7. 8. 9. 10. 11. 12. Rev.S08 Use basic terminology related to inequalities. Use interval notation. Solve linear inequalities symbolically. Solve linear inequalities graphically and numerically. Solve double inequalities. Graph a system of linear inequalities. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 3 System of Equations and Inequalities There are two major topics in this module: - System of Linear Equations in Two Variables System - Solutions of Linear Inequalities Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 4 Do We Really Use Functions of Two Variables? The answer is YES. Many quantities in everyday life depend on more than one variable. Examples Area of a rectangle requires both width and length. Heat index is the function of temperature and humidity. Wind chill is determined by calculating the temperature and wind speed. Grade point average is computed using grades and credit hours. Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 5 Let’s Take a Look at the Arithmetic Operations The arithmetic operations of addition, subtraction, multiplication, and division are computed by functions of two inputs. The addition function of f can be represented symbolically by f(x,y) = x + y, where z = f(x,y). The independent variables are x and y. The dependent variable is z. The z output depends on the inputs x and y. Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 6 Here are Some Examples For each function, evaluate the expression and interpret the result. a) f(5, –2) where f(x,y) = xy b) A(6,9), where calculates the area of a triangle with a base of 6 inches and a height of 9 inches. Solution • f(5, –2) = (5)(–2) = –10. • A(6,9) = If a triangle has a base of 6 inches and a height of 9 inches, the area of the triangle is 27 square inches. Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 7 What is a System of Linear Equations? A linear equation in two variables can be written in the form ax + by = k, where a, b, and k are constants, and a and b are not equal to 0. A pair of equations is called a system of linear equations because they involve solving more than one linear equation at once. A solution to a system of equations consists of an xvalue and a y-value that satisfy both equations simultaneously. The set of all solutions is called the solution set. Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 8 How to Use the Method of Substitution to solve a system of two equations? Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 9 How to Solve the System Symbolically? Solve the system symbolically. Solution Step 1: Solve one of the equations for one of the variables. Rev.S08 Step 2: Substitute for y in the second equation. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 10 How to Solve the System Symbolically? (Cont.) Step 3: Substitute x = 1 into the equation from Step 1. We find that Check: The ordered pair is (1, 2) since the solutions check in both equations. Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 11 Example with Infinitely Many Solutions Solve the system. • Solution • Solve the second equation for y. • Substitute 4x + 2 for y in the first equation, solving for x. • The equation − 4 = − 4 is an identity that is always true and indicates that there are infinitely many solutions. The two equations are equivalent. Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 12 Possible Graphs of a System of Two Linear Equations in Two Variables Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 13 How to Use Elimination Method to Solve System of Equations? Use elimination to solve each system of equations, if possible. Identify the system as consistent or inconsistent. If the system is consistent, state whether the equations are dependent or independent. Support your results graphically. a) 3x − y = 7 5x + y = 9 Rev.S08 b) 5x − y = 8 c) x − y = 5 − 5x + y = − 8 x− y=− 2 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 14 How to Use Elimination Method to Solve System of Equations? (Cont.) Solution a) Eliminate y by adding Eliminate by the equations. Find y by substituting x = 2 in either equation. The solution is (2, − 1). The system is consistent and the equations are independent. Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 15 How to Use Elimination Method to Solve System of Equations? (Cont.) b) If we add the equations we obtain the If following result. following The equation 0 = 0 is an The identity that is always true. always The two equations are equivalent. There are infinitely many solutions. many {(x, y)| 5x − y = 8} Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 16 How to Use Elimination Method to Solve System of Equations? (Cont.) c) If we subtract the second equation from the first, we obtain the following result. The equation 0 = 7 is a The contradiction that is never true. never Therefore there are no solutions, no and the system is inconsistent. system Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 17 Let’s Practice Using Elimination Solve the system by using elimination. Solution Multiply the first equation by 3 and the second equation by 4. Addition eliminates the y-variable. Substituting x = 3 in 2x + 3y = 12 results in 2(3) + 3y = 12 or y = 2 The solution is (3, 2). http://faculty.valenciacc.edu/ashaw/ Rev.S08 Click link to download other modules. 18 Terminology related to Inequalities • Inequalities result whenever the equals sign in Inequalities equals an equation is replaced with any one of the replaced symbols: ≤, ≥, <, > ≤, • Examples of inequalities include: Examples •2x –7 > x +13 2x •x2 ≤ 15 – 21x •xy +9 x < 2x2 xy •35 > 6 35 Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 19 Linear Inequality in One Variable •A linear inequality in one variable is an inequality that can linear be written in the form ax + b > 0 where a ≠ 0. ax (The symbol may be replaced by ≤, ≥, <, > ) ≤, •Examples of linear inequalities in one variable: • 5x + 4 ≤ 2 + 3x simplifies to 2x + 2 ≤ 0 • − 1(x – 3) + 4(2x + 1) > 5 simplifies to 7x + 2 > 0 •Examples of inequalities in one variable which are not Examples linear: linear: • x2 < 1 Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 20 Let’s Look at Interval Notation The solution to a linear inequality in one variable is typically an solution linear interval on the real number line. See examples of interval notation interval See below. below. Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 21 Multiplied by a Negative Number Note that 3 < 5, but if both sides are multiplied by − 1, iin n order to produce a true statement the > symbol must be used. used 3<5 but but − 3>− 5 So when both sides of an inequality are multiplied (or So divided) by a negative number the direction of the inequality must be reversed. inequality Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 22 How to Solve Linear Inequalities Symbolically? The procedure for solving a linear inequality symbolically is the same as the procedure for solving a linear equation, except when both sides except of an inequality are multiplied (or divided) by a negative number the direction of the inequality is reversed. Example of Solving a Linear Equation Symbolically Linear Equation Solve − 2x + 1 = x − 2 Solve − 2x − x = − 2 − 1 − 3x = − 3 x=1 Rev.S08 Example of Solving a Example Linear Inequality Symboliclly Linear Inequality Solve − 2x + 1 < x − 2 Solve − 2x − x < − 2 − 1 − 3x < − 3 x>1 Note that we divided both Note sides by − 3 so the direction so of the inequality was reversed. In interval notation the solution set is (1,∞). the http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 23 How to Solve a Linear Inequality Graphically? Solve Note that the graphs intersect at the point (8.20, 7.59). The graph of y1 is above the graph of y2 to the right of the point of intersection o f is or when x > 8.20. Thus, in interval notation, the solution set is interval (8.20, ∞) (8.20, Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 24 How to Solve a Linear Inequality Numerically? Solve Note that the inequality above becomes y1 ≥ y2 since we let y1 equal the leftequal hand side and y2 equal the right hand side. hand To write the solution set of the inequality we are looking for the values of x in the table for which y1 is the same or larger than y2. Note that when x = − 1.3, y1 1.3, is less than y2; but when x = − 1.4, y1 is larger than y2. By the Intermediate Value Property, there is a value of x between − 1.4 and − 1.3 such that y1 = y2. In order to find an approximation of this value, make a new table in which x is incremented by .01 (note that x is incremented by .1 in the table to the left here.) here.) Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 25 How to Solve a Linear Inequality Numerically? (cont.) Solve Solve To write the solution set of the inequality we are looking for the values To solution of x in the table for which y1 is the same as or larger than y2. Note that when x is approximately − 1.36, y1 equals y2 and when x is smaller than − 1.36 y1 is larger than y2 , so the solutions can be written 1.36 so x ≤ − 1.36 or (− ∞, − 1.36] in interval notation. ∞, Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 26 How to Solve Double Inequalities? • Example: Suppose the Fahrenheit temperature x miles above the ground level is given by T(x) = 88 – 32 x. Determine the altitudes where the 88 Determine air temp is from 300 to 400. air • We must solve the inequality 30 < 88 – 32 x < 40 40 To solve: Isolate the variable x in the middle of the To Isolate the three-part inequality three-part Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 27 How to Solve Double Inequalities? (Cont.) Direction reversed –Divided each side of an inequality by a negative side Thus, between 1.5 and 1.8215 Thus, miles above ground level, the air temperature is between 30 and 40 degrees Fahrenheit. Fahrenheit. Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 28 How to Graph a System of Linear Inequalities? The graph of a linear inequality is a half-plane, which may include the boundary. The boundary line is included when the inequality includes a less than or equal to or greater than or equal to symbol. To determine which part of the plane to shade, select a test point. Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 29 How to Graph a System of Linear Inequalities? (Cont.) Graph the solution set to the inequality x + 4y > 4. Solution Graph the line x + 4y = 4 using a dashed line. Use a test point to determine which half of the plane to shade. Test Point x + 4y > 4 True or False? (4, 2) 4 + 4(2) > 4 True (0, 0) 0 + 4(0) > 4 False Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 30 Example Solve the system of inequalities by shading the solution set. Use the graph to identify one solution. x+y≤3 2x + y ≥ 4 Solution Solve each inequality for y. y ≤ − x + 3 (shade below line) y ≥ − 2x + 4 (shade above line) The point (4, − 2) is a solution. Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 31 What have we learned? We have learned to: 1. 2. 3. 4. 5. 6. Evaluate functions of two variables. Apply the method of substitution. Apply the elimination method. Solve system of equations symbolically. Apply graphical and numerical methods to system of equations. Recognize different types of linear systems. Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 32 What have we learned? (Cont.) 7. 8. 9. 10. 11. 12. Rev.S08 Use basic terminology related to inequalities. Use interval notation. Solve linear inequalities symbolically. Solve linear inequalities graphically and numerically. Solve double inequalities. Graph a system of linear inequalities. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 33 Credit Some of these slides have been adapted/modified in part/whole from the slides of the following textbook: • Rockswold, Gary, Precalculus with Modeling and Visualization, 3th Edition Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 34 ...
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