Getting Started 
Misconception/Error The student is unable to demonstrate that if (a, b) satisfies both P = 0 and Q = 0, then (a, b) must also satisfy Q + kP = 0. 
Examples of Student Work at this Level The student attempts some algebraic manipulations of the equations but is unable to develop a coherent argument. The student may cite properties of equality as justifications but never actually proves the given statement. 
Questions Eliciting Thinking What does it mean that (a, b) is a solution of the system {P = 0 and Q = 0}? In general, what makes an ordered pair of numbers a solution of a system of equations?
How is the equation Q + kP = 0 related to the equations P = 0 and Q = 0?
If (a, b) is a solution of P = 0, would (a, b) also be a solution of kP = 0?
If m = n and s = t, is m + s = n + t? 
Instructional Implications Provide sample expressions for each of P and Q (such as x + y = 7 and x â€“ y = 1). Ensure the student understands that P is equivalent to x + y â€“ 7 and Q is equivalent to x â€“ y â€“ 1. Ask the student whether (5, 2) is a solution of the equations P = 0 and Q = 0. Repeat for (4, 3).
Remind the student what it means for (a, b) to be a solution of the system {P = 0 and Q = 0} and ask the student if either (4, 3) or (5, 2) is a solution of the system {x + y â€“ 7 = 0 and x â€“ y â€“ 1 = 0}. Then, choose a value of k such as k = 5, and ask the student to write the equation of the form Q + kP = 0 using the given examples of P and Q [i.e., (x â€“ y â€“ 1) + 5(x + y â€“ 7) = 0]. Ask the student to determine if (4, 3) is a solution of this equation and then to consider whether (4, 3) is a solution of the system {x + y â€“ 7 = 0 and (x â€“ y â€“ 1) + 5(x + y â€“ 7) = 0}. Ask the student to use the given examples of P and Q to complete a similar exercise for part 2 of the proof.
Assist the student in understanding the goal of the exercise. Explain why it is important to know that replacing one equation with the sum of that equation and a multiple of another produces a system with the same solution. Guide the student to understand that Q + kP = 0 represents the sum of Q = 0 and a multiple of P = 0. Showing that (a, b) is also a solution of the system {P = 0 and Q + kP = 0} allows for the development of strategies (such as â€śeliminationâ€ť) to solve systems of equations.
Review what it means for an ordered pair to be a solution of a system of equations. Remind the student that, by assumption, (a, b) is a solution of the system {P = 0 and Q = 0}, so (a, b) satisfies both P and Q. Showing that (a, b) satisfies Q + kP = 0 means that (a, b) is a solution of the system {P = 0 and Q + kP = 0}. Explain that similar reasoning can be applied to the second part of the proof. Review the properties of equality and ask the student to complete the proof. Explain why both parts of the proof are necessary (i.e., to show that each system has exactly the same set of solutions and neither system has solutions that the other does not have).
Review the properties of equality and ask the student to complete the proof. 
Moving Forward 
Misconception/Error The student does not adequately explain the relationship between (a, b) and the equations in terms of a solution of a system of equations. 
Examples of Student Work at this Level The student shows that if (a, b) satisfies both P = 0 and Q = 0, then (a, b) must also satisfy Q + kP = 0. However, the student does not explain the implications of this for the given systems of equations. For example, the student:Â
 Says if (a, b) satisfies both P = 0 and Q = 0 then (a, b) satisfies kQ = 0 (by the Multiplication Property of Equality). If (a, b) satisfies both P = 0 and kQ = 0, then (a, b) satisfies P + kQ = 0 (by the Addition Property of Equality).
 Reasons similarly for the second part of the proof but never explains why (a, b) satisfies both P = 0 and Q = 0 and neglects to explicitly state that (a, b) is a solution of {P = 0 and Q + kP = 0}.

Questions Eliciting Thinking How do you know (a, b) satisfies both P = 0 and Q = 0?
Did you show that (a, b) is a solution of the system {P = 0 and Q + kP = 0}? 
Instructional Implications Review what it means for an ordered pair to be a solution of a system of equations. Remind the student that, by assumption, (a, b) is a solution of the system {P = 0 and Q = 0} so (a, b) satisfies both P and Q. Showing that (a, b) satisfies Q + kP = 0 means that (a, b) is a solution of the system {P = 0 and Q + kP = 0}. Explain that similar reasoning can be applied to the second part of the proof.
Assist the student in understanding the goal of the exercise. Explain why it is important to know that replacing one equation with the sum of that equation and a multiple of another produces a system with the same solution. Guide the student to understand that Q + kP = 0 represents the sum of Q = 0 and a multiple of P = 0. Showing that (a, b) is also a solution of the system {P = 0 and Q + kP = 0} allows for the development of strategies (such as â€śeliminationâ€ť) to solve systems of equations.
Ensure that the student understands why both parts of the proof are necessary (i.e., to show that each system has exactly the same set of solutions and neither system has solutions that the other does not have).
Allow the student to revise his or her proof to make it more complete. 
Almost There 
Misconception/Error The student is unable to explain why both parts of the exercise are required to prove the statement. 
Examples of Student Work at this Level The student correctly completes the proof but does not understand why both parts are necessary. 
Questions Eliciting Thinking Suppose all of your family is shopping at the mall today. Does that also mean everyone shopping at the mall is in your family? Can you apply this same reasoning to the two parts of the proof? 
Instructional Implications Provide the student with an example to illustrate that legitimate algebraic procedures could lead to a loss of solutions or the introduction of extraneous solutions. For example, the equation Â has twoÂ solutions but dividing both sides by x leads to x â€“ 1 = 0 for which only x = 1 is a solution. This is a case in which a solution is lost, resulting in a difference in the solution sets between the original equation () and the derived equation (x â€“ 1 = 0). Ask the student to explain how it is possible to generate additional solutions by reversing the process in the example (i.e., beginning with x â€“ 1 = 0 and multiplying both sides by x).
Explain that both parts of the proof are necessary in order show that each system has exactly the same set of solutions (i.e., neither system has solutions that the other does not have). 
Got It 
Misconception/Error The student provides complete and correct responses to all components of the task. 
Examples of Student Work at this Level The student is able to complete both parts of the proof:
 Since (a, b) is a solution of the system {P = 0 and Q = 0}, then (a, b) is a solution of both P = 0 and Q = 0. If (a, b) satisfies both P = 0 and Q = 0 then (a, b) satisfies kQ = 0 (by the Multiplication Property of Equality). If (a, b) satisfies both P = 0 and kQ = 0, then (a, b) satisfies P + kQ = 0 (by the Addition Property of Equality). Since (a, b) satisfies both P = 0 and P + kQ = 0, then (a, b) is a solution of the system of equations {P = 0 and Q + kP = 0}.
 Since (a, b) is a solution of the system {P = 0 and Q + kP = 0}, then (a, b) is a solution of both P = 0 and Q + kP = 0. If (a, b) satisfies P = 0, then (a, b) satisfies kP = 0 (by the Multiplication Property of Equality). If (a, b) satisfies both kP = 0 and Q + kP = 0, then (a, b) satisfies Q = 0 (by the Subtraction Property of Equality). Since (a, b) satisfies both P = 0 and Q = 0, then (a, b) is a solution of the system of equations {P = 0 and Q = 0}.
 The student explains that the first part of the proof shows that all of the solutions of {P = 0 and Q = 0} are also solutions of {P = 0 and Q + kP = 0}. The second part of the proof shows that all of the solutions of {P = 0 and Q + kP = 0} are also solutions of {P = 0 and Q = 0}. Both parts are necessary to show that neither system has additional solutions the other does not have.

Questions Eliciting Thinking What procedure is justified by the results of this problem?
Are we assuming that P and Q are linear expressions? 
Instructional Implications Ask the student to consider the generalization to three expressions P, Q, R and address the equivalence of the solution sets of {P = 0, Q = 0, R = 0} and {R = 0, Q + dR = 0, P + kQ + cR = 0}, where c, d, k are nonzero real numbers. 