Solutions of Equations in Two Variables
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1 6.1 Solutions of Equations in Two Variables 6.1 OBJECTIVES 1. Find solutions for an equation in two variables 2. Use ordered pair notation to write solutions for equations in two variables We discussed finding solutions for equations in Chapter 2. Recall that a solution is a value for the variable that satisfies the equation, or makes the equation a true statement. For instance, we know that 4 is a solution of the equation 2x 5 13 We know this is true because, when we replace x with 4, we have A true statement NOTE Recall that an equation is two expressions connected by an equal sign. We now want to consider equations in two variables. An example is x y 5 What will the solution look like? It is not going to be a single number, because there are two variables. Here the solution will be a pair of numbers one value for each of the variables, x and y. Suppose that x has the value 3. In the equation x y 5, you can substitute 3 for x. 3 y 5 Solving for y gives y 2 NOTE An equation in two variables pairs two numbers, one for x and one for y. So the pair of values x 3 and y 2 satisfies the equation because That pair of numbers is then a solution for the equation in two variables. How many such pairs are there? Choose any value for x (or for y). You can always find the other paired or corresponding value in an equation of this form. We say that there are an infinite number of pairs that will satisfy the equation. Each of these pairs is a solution. We will find some other solutions for the equation x y 5 in the following example. Example 1 Solving for Corresponding Values For the equation x y 5, find (a) y if x 5 and (b) x if y 4. (a) If x 5 5 y 5 or y 0 471
2 472 CHAPTER 6 AN INTRODUCTION TO GRAPHING (b) If y 4, x 4 5 or x 1 So the pairs x 5, y 0 and x 1, y 4 are both solutions. CHECK YOURSELF 1 For the equation 2x 3y 26, (a) If x 4, y? (b) If y 0, x? To simplify writing the pairs that satisfy an equation, we use the ordered-pair notation. The numbers are written in parentheses and are separated by a comma. For example, we know that the values x 3andy 2 satisfy the equation x y 5. So we write the pair as CAUTION (3, 2) means x 3 and y 2. (2, 3) means x 2 and y 3. (3, 2) and (2, 3) are entirely different. That s why we call them ordered pairs. The x coordinate (3, 2) The y coordinate The first number of the pair is always the value for x and is called the x coordinate. The second number of the pair is always the value for y and is the y coordinate. Using this ordered-pair notation, we can say that (3, 2), (5, 0), and (1, 4) are all solutions for the equation x y 5. Each pair gives values for x and y that will satisfy the equation. Example 2 Identifying Solutions of Two-Variable Equations Which of the ordered pairs (a) (2, 5), (b) (5, 1), and (c) (3, 4) are solutions for the equation 2x y 9? (a) To check whether (2, 5) is a solution, let x 2 and y 5 and see if the equation is satisfied. NOTE (2, 5) is a solution because a true statement results. 2x y 9 The original equation. x y Substitute 2 for x and 5 for y A true statement (2, 5) is a solution for the equation.
3 SOLUTIONS OF EQUATIONS IN TWO VARIABLES SECTION (b) For (5, 1), let x 5 and y A true statement So (5, 1) is a solution. (c) For (3, 4), let x 3 and y 4. Then Not a true statement So (3, 4) is not a solution for the equation. CHECK YOURSELF 2 Which of the ordered pairs (3, 4), (4, 3), (1, 2), and (0, 5) are solutions for the following equation? 3x y 5 If the equation contains only one variable, then the missing variable can take on any value. Example 3 Identifying Solutions of One-Variable Equations Which of the ordered pairs, (2, 0), (0, 2), (5, 2), (2, 5), and (2, 1) are solutions for the equation x 2? A solution is any ordered pair in which the x coordinate is 2. That makes (2, 0), (2, 5), and (2, 1) solutions for the given equation. CHECK YOURSELF 3 Which of the ordered pairs (3, 0), (0, 3), (3, 3), ( 1, 3), and (3, 1) are solutions for the equation y 3? Remember that, when an ordered pair is presented, the first number is always the x coordinate and the second number is always the y coordinate.
4 474 CHAPTER 6 AN INTRODUCTION TO GRAPHING Example 4 Completing Ordered Pair Solutions Complete the ordered pairs (a) (9, ), (b) (, 1), (c) (0, ), and (d) (, 0) for the equation x 3y 6. NOTE The x coordinate is sometimes called the abscissa and the y coordinate the ordinate. (a) The first number, 9, appearing in (9, ) represents the x value. To complete the pair (9, ), substitute 9 for x and then solve for y. 9 3y 6 3y 3 y 1 (9, 1) is a solution. (b) To complete the pair (, 1), let y be 1 and solve for x. x 3( 1) 6 x 3 6 x 3 (3, 1) is a solution. (c) To complete the pair (0, ), let x be y 6 3y 6 y 2 (0, 2) is a solution. (d) To complete the pair (, 0), let y be 0. x x 0 6 x 6 (6, 0) is a solution. CHECK YOURSELF 4 Complete the ordered pairs below so that each is a solution for the equation 2x 5y 10. (10, ), (, 4), (0, ), and (, 0)
5 SOLUTIONS OF EQUATIONS IN TWO VARIABLES SECTION Example 5 Finding Some Solutions of a Two-Variable Equation Find four solutions for the equation 2x y 8 NOTE Generally, you ll want to pick values for x (or for y) so that the resulting equation in one variable is easy to solve. In this case the values used to form the solutions are up to you.you can assign any value for x (or for y). We ll demonstrate with some possible choices. Solution with x 2: 2 2 y 8 4 y 8 y 4 (2, 4) is a solution. Solution with y 6: 2x 6 8 2x 2 x 1 (1, 6) is a solution. Solution with x 0: NOTE The solutions (0, 8) and (4, 0) will have special significance later in graphing. They are also easy to find! 2 0 y 8 y 8 (0, 8) is a solution. Solution with y 0: 2x 0 8 2x 8 x 4 (4, 0) is a solution. CHECK YOURSELF 5 Find four solutions for x 3y 12.
6 476 CHAPTER 6 AN INTRODUCTION TO GRAPHING CHECK YOURSELF ANSWERS 1. (a) y 6; (b) x (3, 4), (1, 2), and (0, 5) are solutions 3. (0, 3), (3, 3), and ( 1, 3) are solutions 4. (10, 2), ( 5, 4), (0, 2), and (5, 0) 5. (6, 2), (3, 3), (0, 4), and (12, 0) are four possibilities
7 Name 6.1 Exercises Section Date Determine which of the ordered pairs are solutions for the given equation. 1. x y 6 (4, 2), ( 2, 4), (0, 6), ( 3, 9) ANSWERS 2. x y 12 (13, 1), (13, 1), (12, 0), (6, 6) 3. 2x y 8 (5, 2), (4, 0), (0, 8), (6, 4) 4. x 5y 20 (10, 2), (10, 2), (20, 0), (25, 1) 5. 3x y 6 (2, 0), (2, 3), (0, 2), (1, 3) 6. x 2y 8 (8, 0), (0, 4), (5, 1), (10, 1) 7. 2x 3y 6 (0, 2), (3, 0), (6, 2), (0, 2) 8. 8x 4y 16 (2, 0), (6, 8), (0, 4), (6, 6) 2 3, x 2y 12 (4, 0),, (0, 6), 2 3, x 4y 12 ( 4, 0),, (0, 3), 11. y 4x (0, 0), (1, 3), (2, 8), (8, 2) 1 2, 0 5, , y 2x 1 (0, 2), (0, 1),, (3, 5) 13. x 3 (3, 5), (0, 3), (3, 0), (3, 7) 14. y 5 (0, 5), (3, 5), ( 2, 5), (5, 5) Complete the ordered pairs so that each is a solution for the given equation. 15. x y 12 (4, ), (, 5), (0, ), (, 0) x y 7 (, 4), (15, ), (0, ), (, 0) 17. 3x y 9 (3, ), (, 9), (, 3), (0, ) 18. x 5y 20 (0, ), (, 2), (10, ), (, 0) 19. 5x y 15 (, 0), (2, ), (4, ), (, 5) 20. x 3y 9 (0, ), (12, ), (, 0), (, 2) 21. 3x 2y 12 (, 0), (, 6), (2, ), (, 3)
8 ANSWERS x 5y 20 (0, ), (5, ), (, 0), (, 6) 2 3, 23. y 3x 9 (, 0),, (0, ),, x 4y 12 (0, ),, (, 0), 25. y 3x 4 (0, ), (, 5), (, 0), 3 2, 2 3, 8 3, 5 3, 26. y 2x 5 (0, ), (, 5),, (, 1) Find four solutions for each of the following equations. Note: Your answers may vary from those shown in the answer section. 27. x y x y x y x y x 4y x 3y x 5y x 7y y 2x y 8x x y 8 An equation in three variables has an ordered triple as a solution. For example, (1, 2, 2) is a solution to the equation x 2y z 3. Complete the ordered-triple solutions for each equation x y z 0 (2, 3, ) 40. 2x y z 2 (, 1, 3) 41. x y z 0 (1,, 5) 42. x y z 1 (4,, 3) x y z 2 ( 2,, 1) 44. x y z 1 ( 2, 1, ) Hourly wages. When an employee produces x units per hour, the hourly wage in dollars is given by y 0.75x 8. What are the hourly wages for the following number of units: 2, 5, 10, 15, and 20? 46. Temperature conversion. Celsius temperature readings can be converted to Fahrenheit readings using the formula F 9. What is the Fahrenheit 5 C 32 temperature that corresponds to each of the following Celsius temperatures: 10, 0, 15, 100? 478
9 ANSWERS 47. Area. The area of a square is given by A s 2. What is the area of the squares whose sides are 5 centimeters (cm), 10 cm, 12 cm, 15 cm? 48. Unit pricing. When x number of units are sold, the price of each unit (in dollars) is given by p x. Find the unit price when the following quantities are sold: , 7, 9, (a) 49. The number of programs for the disabled in the United States from 1993 to 1997 is approximated by the equation y 162x 4365 in which x is the number of years after Complete the following table 51. (b) x y Your monthly pay as a car salesman is determined using the equation S 200x 1500 in which x is the number of cars you can sell each month. (a) Complete the following table. x S (b) You are offered a job at a salary of $56,400 per year. How many cars would you have to sell per month to equal this salary? 51. You now have had practice solving equations with one variable and equations with two variables. Compare equations with one variable to equations with two variables. How are they alike? How are they different? 52. Each of the following sentences describes pairs of numbers that are related. After completing the sentences in parts (a) to (g), write two of your own sentences in (h) and (i). (a) The number of hours you work determines the amount you are. (b) The number of gallons of gasoline you put in your car determines the amount you. 479
10 ANSWERS a. b. c. d. e. f. g. h. i. j. (c) The amount of the in a restaurant is related to the amount of the tip. (d) The sales amount of a purchase in a store determines. (e) The age of an automobile is related to. (f ) The amount of electricity you use in a month determines. (g) The cost of food for a family of four and. Think of two more: (h). (i). Getting Ready for Section 6.2 [Section 0.4] Plot points with the following coordinates on the number line shown below. (a) 3 (b) 7 (c) 0 (d) 8 (e) Give the coordinate of each of the following points. (f) A (g) B (h) C (i) D (j) E A B C D E Answers 1. (4, 2), (0, 6), ( 3, 9) 3. (5, 2), (4, 0), (6, 4) 5. (2, 0), (1, 3) 7. (3, 0), (6, 2), (0, 2) 2 9. (4, 0), 11. (0, 0), (2, 8) 13. (3, 5), (3, 0), (3, 7) 15. 8, 7, 12, , 0, 4, , 5, 5, , 0, 3, , 11, 9, (0, 7), (2, 5), (4, 3), (6, 1) 29. (0, 6), (3, 0), (6, 6), (9, 12) 31. (8, 0), ( 4, 3), (0, 2), (4, 1) 33. ( 5, 4), (0, 2), (5, 0), (10, 2) 35. (0, 3), (1, 5), (2, 7), (3, 9) 37. ( 5, 0), ( 5, 1), ( 5, 2), ( 5, 3) 39. (2, 3, 1) 41. (1, 6, 5) 43. ( 2, 5, 1) 45. $9.50, $11.75, $15.50, $19.25, $ cm 2, 100 cm 2, 144 cm 2, 225 cm , 4689, 4851, 5013, a e. 10 d a c e b 5 f. 7 g. 4 h. 4 i. 8 j. 3, 5, 5, , 3, 4 3, 1 480
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