Title: M2.1 System of Equations and Inequalities Quiz
Course: Technical Math for Industry > Section: Module 2.1 Systems of Equations and Inequalities

Q: The solution to the system below is .

and

Which graph correctly represents this?
A: (a choice with no text)
Feedback: Incorrect. The line is graphed correctly, but the other line is not. The point of intersection should be the solution to the system, . However, the point of intersection that is pictured is .
Feedback: Incorrect. The line is graphed correctly, but the other line is not. The point of intersection should be the solution to the system, . However, the point of intersection that is pictured is .
Feedback: Incorrect. The line is graphed correctly, but the other line is not. The point of intersection should be the solution to the system, , which it is.
Feedback: Correct. The line goes through and . The line goes through and . These are the lines shown. The point of intersection shown is , which is the solution of the system.

Q: Which system is the only inconsistent system?
A: (a choice with no text)
Feedback: Incorrect. These two equations have the same graph. This means that every point on this line is a solution to the system. If a system has an infinite number of solutions, then it is consistent. A system is inconsistent if there is no solution.
Feedback: Incorrect. The graphs of these two equations are lines that intersect at the point (0, 4). This is a solution of the system, so it is consistent. A system is inconsistent if there is no solution.
Feedback: Incorrect. The graphs of these two equations are lines that intersect at the point (0, 0). This is a solution of the system, so it is consistent. A system is inconsistent if there is no solution.
Feedback: Correct. These two lines have the same slope and different y-intercepts, so they are parallel. Since they do not intersect, the system has no solution. This means that the system is inconsistent. The graph of this system is shown.

Q: Which of the following systems has equations that are dependent?
A: (a choice with no text)
Feedback: Incorrect. You may have confused dependent and independent or graphed the lines incorrectly. The graphs of these equations are lines that intersect in one point, so the equations are independent.
Feedback: Incorrect. You may have confused dependent and independent or graphed the lines incorrectly. The graphs of these equations are parallel lines, so the equations are independent.
Feedback: Correct. The graph of this system is shown below. The single line is the graph of both equations. This means that the equations are dependent.
Feedback: Incorrect. You may have confused dependent and independent or graphed the lines incorrectly. The graphs of these equations are parallel lines, so the equations are independent.

Q: Is a solution to the system and ?
- Yes, because is a solution to one equation. (Incorrect. You may have incorrectly substituted into the equations. The ordered pair is not a solution to either equation and is not a solution to the system. If it were a solution to only one of the equations, then it would still not be a solution to the system.)
- No, because is a solution to neither equation. (Correct. Substitute 4 for x and 2 for y in each equation. Neither equation is true.)
- Yes, because is a solution to both equations. (Incorrect. You may have incorrectly substituted into the equations or made an error with a sign. The ordered pair is not a solution to either equation and it is not a solution to the system. Make sure that you substitute 4 for x and 2 for y, and be careful with your signs.)
- No, because is a solution to only one equation. (Incorrect. While it is true that (4, 2) is not a solution to the system, it is actually not a solution to either of these equations. You may have switched the values of x and y when substituting into the first equation. Make sure that you substitute 4 for x and 2 for y.)
A: No, because is a solution to neither equation.

Q: Lisa and Kate are playing a card game, and a total of 900 points has been scored. Lisa scored 250 more points than Kate. If you let the number of points that Lisa scored, and the number of points that Kate scored, then the problem can be represented by the system:

and

Graph the system. How many points did each of them score?
- Kate = 450 and Lisa = 450 (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 girls.)
- Kate = 250 and Lisa = 650 (Incorrect. You may have chosen this answer because the number 250 was included in the problem. However, 250 is the difference between Lisa and Kate'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 girls.)
- Kate = 575 and Lisa = 325 (Incorrect. You may have drawn the correct graph and 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.)
- Kate = 325 and Lisa = 575 (Correct. The graph of this system is shown. The solution is represented by the point of intersection: . This means that Kate scored 325 points and Lisa scored 575 points.)
A: Kate = 325 and Lisa = 575

Q: The following graph shows the solution to which system of inequalities?
A: (a choice with no text)
Feedback: 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.
Feedback: 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.
Feedback: 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 .
Feedback: 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 (0, 0) a solution of the system and ?
- No, because (0, 0) 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 both inequalities are greater than but not equal to.)
- No, because (0, 0) is not a solution to either inequality. (Correct. When and are substituted into the two inequalities, they produce the false statements and .)
- Yes, because (0, 0) 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.)
- Yes, because (0, 0) is a solution to both inequalities. (Incorrect. You need to substitute the values and into the two inequalities. You may not have noticed that both inequalities are greater than but not equal to.)
A: No, because (0, 0) is not a solution to either inequality.

Q: The profit on the sale of a small bag of popcorn is $1, while the profit on the sale of a medium bag of popcorn is $2. The Glee Club expects to sell fewer than 100 total bags of popcorn, but they hope to make more than $100 in profit. The system that represents this is and . The graph of the boundary lines is shown below. Which regions should be shaded (to form the solution)?
- 1 only (Incorrect. Use a test point in each region. For example, makes false, so region 1 is not shaded.)
- 3 only (Incorrect. Use a test point in each region. For example, makes false, so region 3 is not shaded.)
- 1 and 2 (Incorrect. Use a test point in each region. For example, makes false, so region 1 is not shaded.)
- 2 only (Correct. Use a test point in each region. For example, makes false, so region 1 is not shaded. On the other hand, makes both inequalities true, so region 2 is shaded. Finally, makes false, so region 3 is not shaded.)
A: 2 only

Q: Solve for x and 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 x in the first equation and solve for y. Substitute this value into any equation to find x.)
- , (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 x in the first equation and solve for y. Substitute this value into any equation to find x.)
- , (Correct. Substitute for x in the first equation: Simplify: or Solve for y: , so Substitute this value for y back into the second equation:)
- , (Incorrect. You switched the values of x and y. If you substitute these values into the first equation you get , or , which is false.)
A: ,

Q: Solve for x and y:
- There are an infinite number of solutions. (Incorrect. If you end up with an equation that is always true, such as , then there are an infinite number of solutions. However, if you substitute for x in the second equation and then simplify, you will end up with the equation . Since this is false, you cannot find numbers that make both equations true.)
- The solution is (0, ). (Incorrect. If you solve a system and end up with the equations and , then the solution is (0, ). Here, if you substitute for x in the second equation and then simplify, you will end up with the equation , which means something very different. Since this is false, you cannot find numbers that make both equations true.)
- , (Incorrect. This is a solution to the first equation, but not the second one, so it is not a solution of the system. If you substitute for x in the second equation and then simplify, you will end up with the equation . Since this is false, you cannot find numbers that make both equations true.)
- There is no solution. (Correct. Substitute for x in the second equation: Simplify: or , so Both equations being true would imply that . Since this is false, you cannot find numbers that make both equations true. In other words, the system does not have a solution.)
A: There is no solution.

Q: The school store sells Midville High T-shirts for $12 each and Midville High gym shorts for $9 a pair. One afternoon they sold a total of 53 T-shirts and pairs of shorts for a total of $570. How many T-shirts did they sell and how many pairs of shorts did they sell?
- 34 T-shirts and 18 pairs of shorts (Incorrect. This answer gives the correct total revenue, but not the right number of items sold. There should be 53 T-shirts and pairs of shorts altogether, not 52. The correct equations with which to start the problem are: (A) (B))
- 22 T-shirts and 31 pairs of shorts (Incorrect. You mixed up the number of T-shirts and pairs of shorts. Remember to check your answer against the original information. Twenty-two T-shirts sell for dollars, while 31 pairs of shorts sell for dollars. This would be a total of $543, not $570.)
- 33 T-shirts and 20 pairs of shorts (Incorrect. This answer gives the correct total number of items sold, but not the correct total revenue. Thirty-three T-shirts sell for dollars, while 20 pairs of shorts sell for dollars. This would be a total of $576, not $570. The correct equations with which to start the problem are: (A) (B))
- 31 T-shirts and 22 pairs of shorts (Correct. Let the number of T-shirts sold and let the number of pairs of shorts sold. Then: (A) (B) Solve (A) for t: Substitute into (B): Multiply, and combine like terms: The solution of this equation is , which gives you .)
A: 31 T-shirts and 22 pairs of shorts

Q: Use elimination to solve for x and y:
- (3, 4) (Correct. Add the equations to get: Solve for x to get and substitute into the second equation: , so and)
- (7, 3) (Incorrect. You may have correctly added the equations and gotten , but then mixed up the variables and substituted this value for y in the second equation. 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 11.)
- (4, 3) (Incorrect. You probably solved correctly, but you switched the values of x and y. Remember to check your answer in both equations.)
A: (3, 4)
Feedback: 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 11.

Q: Solve for x and y:
- (7, 5) (Correct. Multiply the first equation by : Add this to to get: Solve for y to get , and substitute this into the first equation: or , so)
- (5, 8) (Incorrect. You may have correctly multiplied the first equation by and combined the equations to find that , but then mixed up the variables. This ordered pair satisfies the second equation, but not the first. If you substitute it into the left side you will get , not .)
A: (7, 5)
Feedback: Incorrect. You may have multiplied the first equation by 3 instead of by 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 .
Feedback: Incorrect. You may have multiplied the first equation by 3 instead of by 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 31.

Q: Solve for x and y:
- There are an infinite number of solutions. (Correct. Multiply the first equation by : Add this to : Since this is true regardless of the value x or y, there are an infinite number of solutions.)
- 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 first equation by and add it to the second equation, then you end up with , which means something very different, and that 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 first equation by and add it to the second equation, then you end up with . This is true regardless of the value of x or y.)
- 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 first equation by and add it to the second 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 coffee beans in one-pound bags. The first mixture has 12 ounces of Sumatra combined with 4 ounces of Celebes Kalossi, and costs $15. The second mixture has 4 ounces of Sumatra and 12 ounces of Celebes Kalossi, and costs $21. How much does one ounce of Sumatra and one ounce of Celebes Kalossi cost?
- $1.65 for Sumatra and $1.20 for Celebes Kalossi (Incorrect. This would give you the correct cost for the second mixture, but not for the first. 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.)
- There is no solution. (Incorrect. You may have mixed up the variables in one of the two equations. You should be starting with the equations and . Multiply the first equation by and add this to the second equation.)
- $1.50 for Sumatra and $0.75 for Celebes Kalossi (Incorrect. You switched the costs of the beans. You should be starting with the two equations: and . Multiply the first equation by and add this to the second equation.)
- $0.75 for Sumatra and $1.50 for Celebes Kalossi (Correct. Let the cost of one ounce of Sumatra and the cost of one ounce of Celebes Kalossi. 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 .)
A: $0.75 for Sumatra and $1.50 for Celebes Kalossi

Q: Solve for x, y, and z.
A: (a choice with no text)
Feedback: Incorrect. You correctly found the value for z to be . 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 y 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.
Feedback: 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 x and z into the second equation: , so Check your solution in all three equations.
Feedback: Incorrect. After eliminating the variables y and x, you made a mistake while solving for z in the equation . You found instead of , and used this value to find the values of x and y. To be a solution, the values must make each of the equations in the system true. These values do not make the third equation true.
Feedback: Incorrect. You correctly found the value for z to be . 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 y 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.

Q: Solve for x, y, and z.
A: (a choice with no text)
Feedback: Incorrect. You correctly found the value of z to be , but later made a mistake to get instead of . You substituted and in the first equation to get . These values do not make the second and third equations true, however.
Feedback: Incorrect. You correctly found the value of z to be , but later made a mistake to get instead of . You substituted and in the third equation to get . These values do not make the first and second equations true, however.
Feedback: Correct. Multiply the first equation by 3: Add this to the second equation to get: Multiply the third equation by 2: Add this to the second equation to get: Multiply the first new equation by 2: Add this to the preceding equation to get: , so From the preceding equation: From an original equation:
Feedback: Incorrect. You may have found the correct values for each variable, but the order of the solution should be , not .

Q: Sasha has some pennies, nickels, and dimes in her pocket. The number of coins is 18. The expression represents the value of the coins, which is $1.14. She has twice as many dimes as pennies. How many of each coin does Sasha have?
- 5 pennies, 3 nickels, and 10 dimes (Incorrect. You may have chosen this answer because the number of dimes is twice the number of pennies and the number of coins is 18. However, to be correct the values need to make each of the conditions true. The correct system to start with is or)
- 4 pennies, 6 nickels, and 8 dimes (Correct. If she has p pennies, n nickels, and d dimes, then: or Multiply the second equation by 100: Multiply the first equation by and add to the above: Multiply the third by 2 and add to this: or From the third equation , and from the first equation .)
- 3 pennies, 9 nickels, and 6 dimes (Incorrect. You may have chosen this answer because the number of dimes is twice the number of pennies and the number of coins is 18. However, to be correct the values need to make each of the conditions true. The correct system to start with is or)
- 2 pennies, 12 nickels, and 4 dimes (Incorrect. You may have chosen this answer because the number of dimes is twice the number of pennies and the number of coins is 18. However, to be correct the values need to make each of the conditions true. The correct system to start with is or)
A: 4 pennies, 6 nickels, and 8 dimes

Q: Which system of equations has no solution?
- ------------------------------ (Correct. If you multiply the third equation by 2 and add it to the first equation, the result is . Because this is a false statement, there is no solution to this system.)
- ------------------------------ (Incorrect. You may have made an error when attempting to eliminate a variable. A system has no solution only if you end up with an equation like . This system has the unique solution .)
- ------------------------------ (Incorrect. When you eliminate the variables in the system, you should get a resulting statement of . This indicates that there are an infinite number of solutions to this system.)
- ------------------------------ (Incorrect. When you eliminate variables for the first and second equations in the system, you should get a resulting statement of . Likewise, when you eliminate variables for the second and third equations, you get the result of . This indicates that there are an infinite number of solutions to this system.)
A: ------------------------------