Getting Started 
Misconception/Error The student does not have an effective strategy for solving systems of linear equations. 
Examples of Student Work at this Level The student:
 Is unable to start the problem.
 Tries to add the equations as they are.
 Tries to solve each equation individually.
 Uses â€śguess and test.â€ť

Questions Eliciting Thinking What is a system of linear equations? What does a solution of a system of equations look like?
What is meant by â€śsolveâ€ť the system?
What are you trying to accomplish by adding the equations together? 
Instructional Implications If needed, review what it means for an ordered pair of numbers to be a solution of a linear equation. Emphasize the onetoone relationship between solutions of linear equations and points on the lines that represent them. Give the student a linear equation such as y = 2x + 3 along with its graph. Ask the student to use the graph to identify several points on the line and then demonstrate that each point satisfies the equation. Next ask the student to identify a point not on the line and use the equation to show that it is not a solution.
Review what it means for an ordered pair to be a solution of a system of linear equations in two variables. Demonstrate finding the solution of a system of two independent equations graphically. Ask the student to use the equations to show that the solution satisfies each equation in the system and is, consequently, a solution of the system. Next, provide instruction on solving systems of linear equations using algebraic methods such as Substitution and Elimination. Provide additional opportunities to use both methods to solve systems of equations. Be sure to include equations with no and infinitely many solutions. Assist the student in recognizing algebraic outcomes associated with these two cases. Eventually, guide the student to consider which method might be better suited to the equations in a particular system. 
Moving Forward 
Misconception/Error The student has a strategy for solving systems, but makes significant algebraic errors. 
Examples of Student Work at this Level The student:
 Combines terms on opposite sides of the equal sign when adding the equations together.
 Uses the wrong operation when attempting to eliminate a variable term from an equation.
 Combines like terms incorrectly.

Questions Eliciting Thinking When using the elimination method, can you eliminate (or combine like terms) if the like terms are on opposite sides of the equal sign? Can you find the error in this step?
What happens when you try to eliminate the same term from each side of an equation? What happens to the 4x on the right side of this equation when you try to eliminate it from the left side of the equation: 4x + 28 = 4x + 10?
Did you check your solutions in the original equations? 
Instructional Implications Review the process for using the elimination and substitution methods. Model the use of the elimination and substitution method for each system and explain the rationale behind each method. Encourage the student to identify which method is better suited to each system. Emphasize the importance of checking the solution in each of the original equations.
Provide the student with guided practice in areas of need. For example, combining like terms, operations with integers, working with equations with variables on both sides of the equation.
Provide additional opportunities to solve systems of equations algebraically and provide feedback. 
Almost There 
Misconception/Error The student has a strategy for solving systems, but makes an easily correctable error. 
Examples of Student Work at this Level The student:
 Makes a sign error.
 Does not understand the significance of the algebraic outcome in the second equation.
 Rewrites part of an equation incorrectly which leads to subsequent errors.
 Checks the solution in only one of the equations.

Questions Eliciting Thinking I think you made an error in this problem. Can you find and fix your error?
For the second system, your work is completely correct but what do you think this means in terms of the solution? 
Instructional Implications Provide feedback to the student concerning the error made and allow the student to revise his or her work. Offer strategies to help the student avoid minor errors. For example, suggest that the student check solutions in both of the original equations and check that each equation was rewritten correctly.
Be sure the student understands the significance of algebraic outcomes when solving systems that result in no or infinitely many solutions. Ask the student to summarize the various possibilities when solving systems of equations in terms of the algebraic outcomes, the nature of the graphs, and the number of solutions. 
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 solves each system correctly getting (4, 9) for the first system, â€śno solutionâ€ť for the second system, and (8, 24) for the third system. For the first and third system, the student checks the solution to see that it satisfies both equations. The student can explain why the second system has no solution. 
Questions Eliciting Thinking Is (4, 9) the only solution for the first system? Why or why not?
What will the graph of each system look like?
Can you predict how many solutions a system will have by looking at the equations (not graphing the equations)?
Is there another way you could have solved each system?
How do you decide which method to use when solving a system? Do you always use the same method? 
Instructional Implications Discuss how many solutions systems represented by intersecting lines, parallel lines, and coinciding lines will have. Present three different systems of linear equations (e.g., one solution, no solution, infinitely many solutions) with each equation written in slopeintercept form. Ask the student to analyze the slopes and yintercepts of the equations in each system and to predict how many solutions each system will have.
Next, present the student with a system of linear equations where the two equations are not written in the same form (e.g., y = 3x â€“ 4 and 3x + y = 2). Show the student how to rewrite the equations for easy comparison. Consider implementing MFAS task How Many Solutions?Â 