Solving Systems

Getting Started

The student does not understand what it means for an ordered pair of numbers to be a solution of a system of equations.

The student is unable to show that (3, 4) is a solution of the given system. The student may attempt to solve the system but is unsuccessful.

How would you determine if (3, 4) is a solution of the equation x + y = 7?

What does it mean for an ordered pair of numbers to be a solution of a system of equations? In general, what makes an ordered pair a solution of a system of equations?

Review what it means for an ordered pair to be a solution of an equation and a solution of a system of equations. Provide systems of equations and ask the student to determine if given ordered pairs are solutions of each system.

Moving Forward

The student is unable to create a new system by replacing one equation with the sum of that equation and a multiple of the other.

The student demonstrates that (3, 4) satisfies both equations in the original system but is unable to correctly write the new system. The student may not understand what is being asked in the second prompt or may make an error when multiplying equation (1) by five or when adding the two equations to form equation (3). 

What does it mean to multiply equation (1) by five? Can you explain how you did this?

What does it mean to add two equations? Can you explain how you added these two equations to form equation (3)?

Explain what it means to multiply an equation by a number and how to add equations. Provide feedback to the student concerning any error made and allow the student to revise his or her work.

Almost There

The student is unable to explain why the two systems have the same solution.

The student demonstrates that (3, 4) satisfies the original system, correctly writes the new system, and shows that (3, 4) also satisfies the new system. However, the student is unable to explain that when one equation in a system is replaced by the sum of that equation and a multiple of the other, the new system formed will still have the same solution as the original system.

Suppose you multiplied equation (1) by -4 instead of five before adding it to equation (2). Would you expect the (3, 4) to be a solution of this system?

What do you actually do when you use the elimination method to solve a system of equations?

Explain that when one equation in a system is replaced by the sum of that equation and a multiple of the other, the new system formed will still have the same solution as the original system. Ask the student to multiply equation (1) by any chosen value (other than five), add it to equation (2) to form equation (3), and then check to see if (3, 4) satisfies the new system consisting of equations (1) and (3).

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 this allows for the development of strategies (such as “elimination”) to solve systems of equations.

Got It

The student provides complete and correct responses to all components of the task.

The student demonstrates that (3, 4) satisfies the original system, correctly writes the new system, and shows that (3, 4) also satisfies the new system. The student explains that when one equation in a system is replaced by the sum of that equation and a multiple of the other, the new system formed will still have the same solution as the original system.

Why is it 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?

Consider implementing MFAS task Solution Sets of Systems to assess the student’s understanding of a more general approach.