Title: Unit 14 Quiz
Course: Technical Math for Industry

Q: Below is the graph of a system of two linear equations.

What system does it represent, and what is the solution of this system?
- and ; solution (Incorrect. The first equation is shown in the graph, so it is part of the system. However, the second equation is not. The solution to the system is the point of intersection and it is correct.)
- and ; solution (Incorrect. The second equation can be rewritten as . These two equations are shown in the graph, so this is the correct system. However, the solution to the system is the point of intersection, and that point is not . Always check your solution in the original equations.)
- and ; solution (Correct. The second equation can be rewritten as . These two equations are shown in the graph. Their point of intersection is . This is the solution to the system because the values and make both equations true: and .)
- and ; solution (Incorrect. The first equation is shown in the graph, so it is part of the system. However, the second equation is not. The solution to the system is the point of intersection, and that point is not . Always check your solution in the original equations.)
A: and ; solution

Q: Which statement about the following system is correct?
- The system is consistent because it has no solution. (Incorrect. You are correct that this system has no solution. However, a consistent system would have at least one solution.)
- The system is inconsistent because it has no solution. (Correct. When the two equations are graphed, the lines are parallel. A system consisting of parallel lines has no solution and a system that has no solution is inconsistent.)
- The system is inconsistent because the two equations actually represent the same line. (Incorrect. You may have incorrectly graphed the equations and obtained the same line for the two equations. If the two equations did represent the same line, there would be an infinite number of solutions, and the system would be consistent. The correct graphs of these equations are two parallel lines.)
- The system is consistent because it has a solution. (Incorrect. You may have incorrectly graphed the equations and obtained either intersecting lines or the same line. If that were correct, the system would be consistent. However, this system does not have a solution.)
A: The system is inconsistent because it has no solution.

Q: Which statement about the following equations is correct?
- The equations are dependent because the lines do not intersect. (Incorrect. You may have incorrectly graphed the equations to show lines that do not intersect, but these lines do intersect. However, you have confused the definitions of dependent and independent. Equations are dependent only if the graphs are the same line.)
- The equations are dependent because the lines are the same line. (Incorrect. You may have incorrectly graphed the two lines or ignored the negative sign on the second equation's x-coefficient. They are not the same line. The first thing to note is the slopes of the two lines and what this means about their relationship. The second thing to note is their y-intercepts. This will help you graph them.)
- The equations are independent because the equations represent parallel lines. (Incorrect. You may have incorrectly graphed the two lines. Correctly graphed, the lines intersect in one point. The first thing to note is the slopes of the two lines and what this means about their relationship. Then note their y-intercepts. This will help you graph them.)
- The equations are independent because the lines intersect in one point. (Correct. The lines have different slopes and when graphed intersect in one point . This means that they are distinct lines, so the equations are independent. The lines are graphed below.)
A: The equations are independent because the lines intersect in one point.

Q: Which statement is true about the system and ?
- is a solution to one equation but not the other one, so it is a solution to the system. (Incorrect. An ordered pair is a solution to a system only if it makes both equations true. Check your calculations again to see to which equations is a solution.)
- is not a solution to either equation, so it is not a solution to the system. (Incorrect. You may have incorrectly substituted into both of the equations. Make sure that you replace x by 8 and y by 1 in both equations, and check your calculations.)
- is a solution to both equations, so it is a solution to the system. (Correct. Substitute 8 for x and 1 for y. | True True | Since the ordered pair makes both equations true, it is a solution to the system.)
- is a solution to one equation but not the other one, so it is not a solution to the system. (Incorrect. You may have incorrectly substituted into one of the equations. Make sure that you replace x by 8 and y by 1 in both equations, and check your calculations.)
A: is a solution to both equations, so it is a solution to the system.

Q: Luca and William are playing a video game and they have scored a total of 20,000 points. Luca scored 4,000 more points than William. If you let the number of points that Luca scored, and the number of points that William scored, then the problem can be represented by the system:

and

Graph the system. How many points did each boy score?
- William = 12,000 and Luca = 8,000 (Incorrect. You may have drawn the correct graph, but then misread it. The solution is represented by the point of intersection. Make sure that you carefully note which person is represented by the horizontal axis and which person is represented by the vertical axis.)
- William = 8,000 and Luca = 12,000 (Correct. The graph of this system is shown. The solution is represented by the point of intersection: . This means that Luca scored 12,000 points and William scored 8,000 points.)
- William = 4,000 and Luca = 16,000 (Incorrect. You may have chosen this answer because the number 4,000 was included in the problem. However, 4,000 is the difference between Luca's and William's scores. Graph the two equations. The solution is represented by the point of intersection. The two coordinates will be the amounts scored by the two boys.)
- William = 10,000 and Luca = 10,000 (Incorrect. These numbers are a solution to one equation, but not to both. Graph the two equations. The solution is represented by the point of intersection. The two coordinates will be the amounts scored by the two boys.)
A: William = 8,000 and Luca = 12,000

Q: The following graph shows the solution to which system of inequalities?
A: (a choice with no text)
If wrong: Incorrect. The boundary lines , , and are shown, and the solution to this system is contained in the shaded region. However, you may have confused the use of solid and dashed lines. A solid line is drawn when the inequality uses or .
If wrong: Incorrect. The boundary lines and are shown, but the shaded region is also bounded by . Besides this, the points in the shaded region do not satisfy . Be sure to test a point from the shaded region in each inequality.
If wrong: Incorrect. The boundary lines , , and are shown. However, the points in the shaded region do not satisfy . Be sure to test a point from the shaded region in each inequality.
If right: Correct. The boundary lines , , and are shown. Test a point inside the region to see if it makes these inequalities true. For example, the point is inside the region. Substituting in the first inequality gives you , which is true. Substituting in the second inequality gives you , or , which is true.

Q: Is (1, 2) a solution of the system and ?
- No, because (1, 2) is not a solution to either inequality. (Correct. When and are substituted into the two inequalities, they produce the false statements and .)
- Yes, because (1, 2) is a solution to both inequalities. (Incorrect. You need to substitute the values and into the two inequalities. You may not have noticed that the inequalities are greater or less than but not equal to.)
- Yes, because (1, 2) is the point of intersection of the two boundary lines of this system. (Incorrect. You may have been thinking about the solution to a system of equations. An ordered pair is a solution to a system of inequalities if it satisfies both inequalities. You need to substitute the values and into the two inequalities to see if they are true.)
- No, because (1, 2) is a solution to only one of these inequalities. (Incorrect. You need to substitute the values and into the two inequalities. You may not have noticed that the inequalities are greater or less than but not equal to.)
A: No, because (1, 2) is not a solution to either inequality.

Q: The profit on the sale of a bicycle at a bike shop is $30, while the profit on the sale of a tricycle is $20. The owner is interested in making more than $200 in profit per day, and expects to sell at most 9 items per day. The system that represents this situation is and . The graph of the boundary lines is shown below. Which region should be shaded (to form the solution)?
- 3 (Correct. Use a test point in each region. Region 1 does not contain any whole number points to test, so region 1 is not shaded. Point makes false, so region 2 is not shaded. Point makes both inequalities true, so region 3 is shaded, while point (1, 1) makes false, so region 4 is not shaded.)
- 4 (Incorrect. Use a test point in each region. For example, point (1, 1) makes false, so region 4 is not shaded.)
- 2 (Incorrect. Use a test point in each region. For example, point makes false, so region 2 is not shaded.)
- 1 (Incorrect. Use a test point in each region. Region 1 does not contain any whole number points to test, so region 1 is not shaded.)
A: 3

Q: Solve for x and y:
- , (Incorrect. You switched the values of x and y. If you substitute these values into the first equation you get , or , which is false.)
- , (Correct. Substitute for y in the second equation: Simplify: or Solve for x: , so Substitute this value for x back into the first equation:)
- , (Incorrect. This is a solution to the second equation, but not to the system. If you substitute these values into the first equation you get , or , which is false. Substitute for y in the second equation and solve for x. Substitute this value into any equation to find y.)
- , (Incorrect. This is a solution to the first equation, but not to the system. If you substitute these values into the second equation you get , or , which is false. Substitute for y in the second equation and solve for x. Substitute this value into any equation to find y.)
A: ,

Q: Solve for x and y:
- , (Incorrect. This is one solution to the system, but not the only one. If you substitute for y in the first equation and then simplify, you will end up with the equation . This equation is true regardless of the value of x.)
- There are an infinite number of solutions. (Correct. Substitute for y in the first equation: Simplify: or Since this equation is true no matter what value x is, this means that every value of x will give you a solution. So there are an infinite number of solutions.)
- The solution is (15, 15). (Incorrect. If you solve a system and end up with the equations and , then the solution is . Here, if you substitute for y in the first equation and then simplify, you will end up with the equation . This equation is true regardless of the value of x.)
- There is no solution. (Incorrect. If you had a system that led to an impossible equation, such as , then there would be no solution. If you substitute for y in the first equation and then simplify, you will end up with the equation . This equation is true regardless of the value of x.)
A: There are an infinite number of solutions.

Q: A bakery sold apple pies for $11 and blueberry pies for $13. One Saturday they sold a total of 38 pies and collected a total of $460. How many apple pies did they sell and how many blueberry pies did they sell?
- 21 apple pies and 17 blueberry pies (Incorrect. You mixed up the number of apple pies and blueberry pies. Remember to check your answer against the original information. Twenty-one apple pies sell for dollars, while 17 blueberry pies sell for dollars. This would be a total of $452, not $460.)
- 19 apple pies and 19 blueberry pies (Incorrect. This answer gives the correct total number of items sold, but not the correct total revenue. Nineteen apple pies sell for dollars, while 19 blueberry pies sell for dollars. This would be a total of $456, not $460. The correct equations with which to start the problem are: (A) (B))
- 4 apple pies and 32 blueberry pies (Incorrect. This answer gives the correct total revenue, but not the right number of items sold. There should be 38 pies altogether, not 36. The correct equations with which to start the problem are: (A) (B))
- 17 apple pies and 21 blueberry pies (Correct. Let the number of apple pies sold and let the number of blueberry pies sold. Then: (A) (B) Solve (A) for a: Substitute into (B): Multiply, and combine like terms: The solution of this equation is , which gives you .)
A: 17 apple pies and 21 blueberry pies

Q: Use elimination to solve for x and y:
A: (a choice with no text)
If right: Correct. Add the equations to get: Solve for y to get and substitute into the first equation: , so and
If wrong: Incorrect. You probably solved correctly, but you switched the values of x and y. Remember to check your answer in both equations.
If wrong: Incorrect. You may have correctly added the equations and gotten , but then mixed up the variables and substituted this value for x in the first equation. This ordered pair is a solution for the first equation, but not for the second equation. If you substituted into the left side you would get , not 1.
If wrong: Incorrect. You probably thought that you had to find the opposite of the second equation before adding the equations. This ordered pair is a solution for the second equation, but not for the first equation. If you substituted into the left side you would get , not 9.

Q: Solve for x and y:
A: (a choice with no text)
If right: Correct. Multiply the first equation by 2: Add this to to get: Solve for x to get , and substitute this into the first equation: or , so and
If wrong: Incorrect. You may have correctly multiplied the first equation by 2, combined the equations, and found the correct values, but then mixed up the variables. Always remember to check your answer in the original equations.
If wrong: Incorrect. You may have multiplied the first equation by instead of by 2 and made a mistake combining it with the second equation. This ordered pair satisfies the second equation, but not the first. If you substitute it into the left side you will get , not 11.
If wrong: Incorrect. You may have multiplied the first equation by instead of by 2 and made a mistake combining it with the second equation. This ordered pair satisfies the first equation, but not the second. If you substitute it into the left side you will get , not 6.

Q: Solve for x and y:
- There are an infinite number of solutions. (Correct. Multiply the second equation by : Add this to : Since this is true regardless of the value of x or y, there are an infinite number of solutions.)
- There is no solution. (Incorrect. If you had a system that led to an impossible equation, such as , then there would be no solution. If you multiply the second equation by and add it to the first equation, then you end up with . This is true regardless of the value of x or y.)
- , (Incorrect. This is one solution to the system, but not the only one. If you multiply the second equation by and add it to the first equation, then you end up with . This is true regardless of the value of x or y.)
- The solution is (0, 0). (Incorrect. If you solve a system and end up with the equations and , then the solution is (0, 0). Here, if you multiply the second equation by and add it to the first equation, then you end up with . This is true regardless of the value of x or y.)
A: There are an infinite number of solutions.

Q: A store is selling two mixtures of nuts in 20-ounce bags. The first mixture has 15 ounces of peanuts combined with five ounces of cashews, and costs $4.25. The second mixture has five ounces of peanuts and 15 ounces of cashews, and costs $6.75. How much does one ounce of peanuts and one ounce of cashews cost?
- There is no solution. (Incorrect. You may have mixed up the variables in one of the two equations. You should be starting with the two equations: and . Multiply the first equation by and add this to the second equation.)
- $0.12 for peanuts and $0.49 for cashews (Incorrect. This would give you the correct cost for the first mixture, but not for the second. You probably made an addition error when combining the two equations to eliminate a variable. You should be starting with the two equations: and . Multiply the first equation by and add this to the second equation.)
- $0.15 for peanuts and $0.40 for cashews (Correct. Let the cost of one ounce of peanuts and the cost of one ounce of cashews. Then the information leads you to two equations: and Multiply the first equation by : Add this to the second equation: , so Substitute into the first equation: or This gives you , so .)
- $0.40 for peanuts and $0.15 for cashews (Incorrect. You switched the costs of the nuts. You should be starting with the two equations: and . Multiply the first equation by and add this to the second equation.)
A: $0.15 for peanuts and $0.40 for cashews

Q: Solve for x, y, and z.
A: (a choice with no text)
If wrong: Incorrect. You correctly found the value for y to be 3. However, you may have made a mistake when substituting this value into , and found instead of . You then used the values and to solve for x in the first equation. To be a solution, the values must make each of the equations in the system true. However, these values do not make the second or third equation true.
If right: Correct. Add the first and second equations: Add the first and third equations: Add the two new equations: , so Substitute this into the first new equation: , so Substitute y and z into the third equation: , so Check your solution in all three equations.
If wrong: Incorrect. You correctly found the value for y to be 3. However, you may have made a mistake when substituting this value into , and found instead of . You then used the values and to solve for x in the first equation. To be a solution, the values must make each of the equations in the system true. However, these values do not make the second or third equation true.
If wrong: Incorrect. After finding the correct values for y and z, you made a mistake while solving for x in the first equation, . To be a solution, the values must make each of the equations in the system true. However, these values do not make any of the equations true.

Q: Solve for x, y, and z.
A: (a choice with no text)
If wrong: Incorrect. You correctly found the value of x to be , but later made a mistake to get instead of . You substituted and in an original equation to get . These values do not make the first and third equations true, however.
If wrong: Incorrect. You made a mistake when solving for x and incorrectly found the value of x to be instead of . You used this value to solve for z in a previous equation to get . You substituted and in the first equation to get . These values do not make the third equation true, however.
If wrong: Incorrect. You may have found the correct values for each variable, but the order of the solution should be , not .
If right: Correct. Multiply the second equation by 2: Add this to the first equation to get: Multiply the first equation by 2: Add this to the third equation to get: Add this to the preceding equation with x and z to get: , so From the preceding equation: From an original equation:

Q: Jessie has some nickels, dimes, and quarters in her bank. The number of coins is 31. The expression represents the value of the coins, which is $3.25. Jessie has four more dimes than she does quarters. How many of each coin does Jessie have?
- 17 nickels, 9 dimes, and 5 quarters (Incorrect. You may have chosen this answer because there are four more dimes than quarters and the number of coins is 31. However, to be correct the values need to make each of the conditions true. The correct system to start with is or)
- 19 nickels, 8 dimes, and 4 quarters (Incorrect. You may have chosen this answer because there are four more dimes than quarters and the number of coins is 31. However, to be correct the values need to make each of the conditions true. The correct system to start with is or)
- 15 nickels, 10 dimes, and 6 quarters (Correct. If Jessie has n nickels, d dimes, and q quarters, then: or Multiply the second equation by 100: Multiply the first equation by and add to the above: Multiply the third by and add to this: or From the third equation: , and from the first equation:)
- 13 nickels, 11 dimes, and 7 quarters (Incorrect. You may have chosen this answer because there are four more dimes than quarters and the number of coins is 31. However, to be correct the values need to make each of the conditions true. The correct system to start with is or)
A: 15 nickels, 10 dimes, and 6 quarters

Q: How many solutions does this system have?
- There is no solution. (Incorrect. When you attempted to eliminate variables, you may have arrived at and interpreted the zeros to mean that there is no solution. However, it is only when you end up with a false statement, such as , that you can conclude that the system has no solution.)
- There is one solution. (Incorrect. You may have thought that a system of equations should have one solution. However, when attempting to eliminate variables, you should get the statement . If there were one solution you would not get an equation without any variables.)
- There are three solutions. (Incorrect. You may have thought that the system of equations should have three solutions because there are three equations, but this is not possible. When attempting to eliminate variables, you will either end up with a solution, or an equation with no variables. In the second case, there is either no solution or an infinite number of solutions.)
- There are an infinite number of solutions. (Correct. If you multiply the third equation by and add it to the first equation, the resulting statement is . If you multiply the third equation by 4 and add it to the second equation, the resulting statement is also . This means that the system has an infinite number of solutions.)
A: There are an infinite number of solutions.