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Study Guide and Review - Chapter 6 State whether each sentence is true or false. If false, replace the underlined term to make a true sentence. 1. If a system has at least one solution, it is said to be consistent. SOLUTION: If a system has at least one solution, it is said to be consistent. So, the statement is true. 2. If a consistent system has exactly two solution(s), it is said to be independent. SOLUTION: The statement is false. If a consistent system has exactly one solution(s), it is said to be independent. 3. If a consistent system has an infinite number of solutions, it is said to be inconsistent. SOLUTION: The statement is false. If a consistent system has an infinite number of solutions, it is said to be dependent. 4. If a system has no solution, it is said to be inconsistent. SOLUTION: If a system has no solution, it is said to be inconsistent. So, the statement is true. 5. Substitution involves substituting an expression from one equation for a variable in the other. SOLUTION: Substitution involves substituting an expression from one equation for a variable in the other. So, the statement is true. 6. In some cases, dividing two equations in a system together will eliminate one of the variables. This process is called elimination. SOLUTION: The statement is false. In some cases, when adding or subtracting two equations in a system together will eliminate one of the variables, this process is called elimination. 7. A set of two or more inequalities with the same variables is called a system of equations. SOLUTION: The statement is false. A set of two or more inequalities with the same variables is called a system of inequalities. 8. When the graphs of the inequalities in a system of inequalities do not intersect, there are no solutions to the system. SOLUTION: True Graph each system and determine the number of solutions that it has. If it has one solution, name it. 9. x − y = 1 x +y = 5 SOLUTION: To graph the system, write both equations in slope-intercept form. Equation 1: eSolutions Manual - Powered by Cognero Page 1 The statement is false. A set of two or more inequalities with the same variables is called a system of inequalities. 8. When the graphs of the inequalities in a system of inequalities do not intersect, there are no solutions to the system. SOLUTION: Study Guide and Review - Chapter 6 True Graph each system and determine the number of solutions that it has. If it has one solution, name it. 9. x − y = 1 x +y = 5 SOLUTION: To graph the system, write both equations in slope-intercept form. Equation 1: Equation 2: Graph and locate the solution. y=x−1 y = −x + 5 The graphs appear to intersect at the point (3, 2). You can check this by substituting 3 for x and 2 for y. The solution is (3, 2). 12. −3x + y = −3 y =x−3 SOLUTION: To graph the system, write both equations in slope-intercept form. eSolutions Manual - Powered by Cognero Equation 1: Page 2 Study Guide andisReview The solution (3, 2). - Chapter 6 12. −3x + y = −3 y =x−3 SOLUTION: To graph the system, write both equations in slope-intercept form. Equation 1: Graph and find the solution. y = 3x − 3 y=x−3 The graphs appear to intersect at the point (0, −3). You can check this by substituting 0 for x and −3 for y. The solution is (0, −3). 15. MAGIC NUMBERS Sean is trying to find two numbers with a sum of 14 and a difference of 4. Define two variables, write a system of equations, and solve by graphing. SOLUTION: Sample answer: Let x be one number and y be the other number. x + y = 14 x −y = 4 To graph the system, write both equations in slope-intercept form. eSolutions Manual - Powered by Cognero Equation 1: Page 3 Guide and Review - Chapter 6 Study The solution is (0, −3). 15. MAGIC NUMBERS Sean is trying to find two numbers with a sum of 14 and a difference of 4. Define two variables, write a system of equations, and solve by graphing. SOLUTION: Sample answer: Let x be one number and y be the other number. x + y = 14 x −y = 4 To graph the system, write both equations in slope-intercept form. Equation 1: Equation 2: Graph the equations and find the solution. y = −x + 14 y=x−4 The graphs appear to intersect at the point (9, 5). So, the numbers 9 and 5 have a sum of 14 and a difference of 4. Use substitution to solve each system of equations. 16. x + y = 3 x = 2y eSolutions Manual - Powered by Cognero SOLUTION: x +y = 3 Page 4 Guide and Review - Chapter 6 Study The graphs appear to intersect at the point (9, 5). So, the numbers 9 and 5 have a sum of 14 and a difference of 4. Use substitution to solve each system of equations. 16. x + y = 3 x = 2y SOLUTION: x +y = 3 x = 2y Substitute 2y for x in the first equation. Use the solution for y and either equation to find x. x = 2y x = 2(1) x =2 The solution is (2, 1). 18. 3x + 2y = 16 x = 3y − 2 SOLUTION: 3x + 2y = 16 x = 3y − 2 Substitute 3y − 2 for x in the first equation. Use the solution for y and either equation to find x. The solution is (4, 2). eSolutions Manual = 9 - Powered by Cognero 21. x + 3y x +y = 1 Page 5 Guide and Review - Chapter 6 Study The solution is (4, 2). 21. x + 3y = 9 x +y = 1 SOLUTION: x + 3y = 9 x +y = 1 First, solve the second equation for y to get y = −x +1. Then substitute −x + 1 for y in the first equation. Use the solution for x and either equation to find y. The solution is (−3, 4). 22. GEOMETRY The perimeter of a rectangle is 48 inches. The length is 6 inches greater than the width. Define the variables, and write equations to represent this situation. Solve the system by using substitution. SOLUTION: Sample answer: Let w be the width and be the length. 2 + 2w = 48 =w+6 Substitute w + 6 for in the first equation. Use the solution for w and either equation to find . =w+6 =9+6 = 15 The solution is (9, 15). Use Manual elimination each eSolutions - Poweredto bysolve Cognero 24. −3x + 4y = 21 3x + 3y = 14 system of equations. Page 6 =w+6 =9+6 = 15 Study Guide and Review - Chapter 6 The solution is (9, 15). Use elimination to solve each system of equations. 24. −3x + 4y = 21 3x + 3y = 14 SOLUTION: Because −3x and 3x have opposite coefficients, add the equations. Now, substitute 5 for y in either equation to find x. The solution is . 27. 6x + y = 9 −6x + 3y = 15 SOLUTION: Because 6x and −6x have opposite coefficients, add the equations. Now, substitute 6 for y in either equation to find x. The solution is . eSolutions Manual - Powered by Cognero 30. 3x + 2y = 8 x + 2y = 2 Page 7 The solution Study Guide andisReview - .Chapter 6 27. 6x + y = 9 −6x + 3y = 15 SOLUTION: Because 6x and −6x have opposite coefficients, add the equations. Now, substitute 6 for y in either equation to find x. The solution is . 30. 3x + 2y = 8 x + 2y = 2 SOLUTION: Because 2y and 2y have same coefficients, multiply equation 2 by –1 so the terms are additive inverses. Then add the equations. Now, substitute 3 for x in either equation to find y. eSolutions Manual - Powered by Cognero The solution is . Page 8 Study Guide andisReview .- Chapter 6 The solution 30. 3x + 2y = 8 x + 2y = 2 SOLUTION: Because 2y and 2y have same coefficients, multiply equation 2 by –1 so the terms are additive inverses. Then add the equations. Now, substitute 3 for x in either equation to find y. The solution is . Use elimination to solve each system of equations. 33. x − y = −2 2x + 4y = 38 SOLUTION: Notice that if you multiply the first equation by 4, the coefficients of the y-terms are additive inverses. Now, substitute 5 for x in either equation to find y. The Manual solution is (5, 7). eSolutions - Powered by Cognero 36. 8x − 3y = −35 Page 9 Study Guide andisReview - .Chapter 6 The solution Use elimination to solve each system of equations. 33. x − y = −2 2x + 4y = 38 SOLUTION: Notice that if you multiply the first equation by 4, the coefficients of the y-terms are additive inverses. Now, substitute 5 for x in either equation to find y. The solution is (5, 7). 36. 8x − 3y = −35 3x + 4y = 33 SOLUTION: Notice that if you multiply the first equation by 4 and the second equation by 3, the coefficients of the y-terms are additive inverses. Now, substitute −1 for x in either equation to find y. The solution is (−1, 9). 39. 8x − 5y = 18 6x + 6y = −6 SOLUTION: Notice that if you multiply the first equation by 6 and the second equation by 5, the coefficients of the y-terms are eSolutions Manual - Powered by Cognero Page 10 additive inverses. Study Guide and Review - Chapter 6 The solution is (−1, 9). 39. 8x − 5y = 18 6x + 6y = −6 SOLUTION: Notice that if you multiply the first equation by 6 and the second equation by 5, the coefficients of the y-terms are additive inverses. Now, substitute 1 for x in either equation to find y. The solution is (1, −2). Determine the best method to solve each system of equations. Then solve the system. 42. y = −x y = 2x SOLUTION: Because both equations are solved for one of the variables, substitution is the best method. Substitute 2x for y in the first equation. Substitute 0 for x in either equation to find y. The solution is (0, 0). 45. 3x + 2y = −4 5x + 2y = −8 SOLUTION: Because 2y and 2y have the same coefficient, elimination using addition is the best method. Multiply equation 2 by – eSolutions Page 11 1. Manual - Powered by Cognero Study Guide andisReview The solution (0, 0). - Chapter 6 45. 3x + 2y = −4 5x + 2y = −8 SOLUTION: Because 2y and 2y have the same coefficient, elimination using addition is the best method. Multiply equation 2 by – 1. Now, substitute −2 for x in either equation to find y. The solution is (−2, 1). 48. 11x − 6y = 3 5x − 8y = −25 SOLUTION: Because none of the coefficients are 1 or −1, elimination using multiplication is the best method. Notice that if you multiply the first equation by 4 and the second equation by −3, the coefficients of the y-terms are additive inverses. Now, substitute 3 for x in either equation to find y. eSolutions Manual - Powered by Cognero The solution is (3, 5). Page 12 Study Guide and Review - Chapter 6 The solution is (−2, 1). 48. 11x − 6y = 3 5x − 8y = −25 SOLUTION: Because none of the coefficients are 1 or −1, elimination using multiplication is the best method. Notice that if you multiply the first equation by 4 and the second equation by −3, the coefficients of the y-terms are additive inverses. Now, substitute 3 for x in either equation to find y. The solution is (3, 5). Solve each system of inequalities by graphing. 51. x > 3 y <x+2 SOLUTION: Graph each inequality. The graph of x > 3 is dashed and is not included in the graph of the solution. The graph of y < x + 2 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of x > 3 and y < x + 2. Overlay Page 13 the graphs and locate the green region. This is the intersection. eSolutions Manual - Powered by Cognero Study Guide and Review - Chapter 6 The solution is (3, 5). Solve each system of inequalities by graphing. 51. x > 3 y <x+2 SOLUTION: Graph each inequality. The graph of x > 3 is dashed and is not included in the graph of the solution. The graph of y < x + 2 is also dashed and is not included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of x > 3 and y < x + 2. Overlay the graphs and locate the green region. This is the intersection. The solution region is shaded in the graph below. 54. y ≤ −x − 3 y ≥ 3x − 2 eSolutions Manual - Powered by Cognero SOLUTION: Graph each inequality. Page 14 Study Guide and Review - Chapter 6 54. y ≤ −x − 3 y ≥ 3x − 2 SOLUTION: Graph each inequality. The graph of y ≤ −x − 3 is solid and is included in the graph of the solution. The graph of y ≥ 3x − 2 is also solid and is included in the graph of the solution. The solution of the system is the set of ordered pairs in the intersection of the graphs of y ≤ −x − 3 and y ≥ 3x − 2. Overlay the graphs and locate the green region. This is the intersection. The solution region is shaded in the graph below. eSolutions Manual - Powered by Cognero Page 15