Getting Started 
Misconception/Error The student does not have an effective strategy for solving a system of linear equations. 
Examples of Student Work at this Level The student:
 Adds the two given equations getting a third equation in two variables.
 Substitutes zero for one of the variables and solves for the other variable in each equation, as if finding the intercepts.
 Eliminates a variable term by subtracting the term from one side of the equation but only its coefficient from the other side.
 Attempts to solve each equation individually for each of the variables.
 Attempts a guessandcheck approach.
 Tries several algebraic operations without a clear plan.

Questions Eliciting Thinking What does a solution of a system of equations look like? What is meant by “solve” the system?
Do you know of any methods that can be used to solve a system of equations?
Which variable are you trying to eliminate? How can you eliminate one of the variables? 
Instructional Implications Review what it means for an ordered pair to be a solution of a system of linear equations in two variables. Provide an example of a system of equations and ask the student to show that the solution satisfies each equation in 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. Address common errors as they occur, (e.g., combining unlike terms, not distributing correctly, adding unequal quantities to both sides of the equation, and computation errors when working with integers and rational numbers). 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 of equations, but makes significant errors. 
Examples of Student Work at this Level The student:
 Chooses the wrong number to multiply by when attempting a linear combination/elimination method.
 Makes distribution errors.
 Adds rather than subtracts terms (i.e., saying 32s + 35s = 67s).
 Solves one equation for g and the other equation for s in an incorrect attempt to use a substitution method.
 Solves one equation for a variable and then substitutes the resulting expression into that same equation.

Questions Eliciting Thinking How did you decide which number to use when multiplying each equation? What are you trying to achieve by multiplying the equations?
In this step, were you adding or subtracting these equations? How did you decide which to do?
By substituting an expression into the same equation you used to solve for a variable you ended up with 0 = 0. What happened to the variable? What does 0 = 0 mean? Can you substitute that expression into the other equation and solve again? 
Instructional Implications Review the processes for using the elimination and substitution methods to solve systems of equations. Model the use of the student’s choice of method for the system explaining each step. Provide feedback to the student with regard to any error(s) made and allow the student to revise incorrect work. 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, using the Distributive Property, and operations with integers and rational numbers.
Provide additional opportunities to solve systems of equations algebraically, and provide feedback. 
Almost There 
Misconception/Error The student makes minor errors while solving pairs of simultaneous linear equations. 
Examples of Student Work at this Level The student:
 Makes a minor calculation or sign error in some step of the problem.
 Copies the problem incorrectly or misreads prior written work, (e.g., reads the variable s as the number five).
 Rounds a solution unnecessarily.
 Writes the answer as an ordered pair or as individual values without associated units or context.
 Misuses the dollar or cent sign, writing .99$ or 0.99¢
 Neglects to explicitly answer the questions asked in the problem.

Questions Eliciting Thinking How could you check your work to see if your answers are correct? Did you check your solutions in both original equations?
Why did you round your first answer before substituting it into the second equation? Do you think this affected your final answer? When is it appropriate to round?
What does your answer mean in relation to the original word problem? What units are appropriate? 
Instructional Implications Provide feedback to the student concerning the error made and allow the student to revise incorrect work. Offer strategies to help the student avoid minor errors. For example, suggest that the student check that each equation was rewritten correctly when multiplying it by a value and check solutions in both of the original equations. Guide the student to explicitly answer any question asked in the problem.
Note: The System Solutions worksheet is editable and can be rewritten with new values and/or a different context to give the student further practice. 
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 correctly solves the system of equations and states that songs cost $0.99 each and games cost $2.99 each.

Questions Eliciting Thinking Is there a different way you could have solved this system of equations? Why did you choose the method that you used?
Would solving by graphing be a good method for this problem? Why or why not? 
Instructional Implications Provide the student with opportunities to solve a wide range of linear systems including those in which different strategies are appropriate. Discuss with the student when using a substitution strategy might be better than using an elimination strategy. Provide additional opportunities to use both strategies 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. Also include equations written in standard form, slopeintercept form and a combination of forms. Consider implementing MFAS task How Many Solutions? (8.EE.3.8).
Expose the student to a wide variety of contexts and structures for writing systems of equations, such as mixture problems, rate problems, investment comparisons, work ratios, and geometry contexts. Ask the student to both write and solve systems of equations. Consider using MFAS task Writing System Equations (8.EE.3.8). 