CCSS.Math.Content.HSA.REI.C.5

Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.

CCSS.Math.Content.HSA.REI.C.6

Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.

Within this task students will need to create multiple models of word problems that define a system of equations. Students will need to algebraically and graphically model the problems to solve them. In having students work through these problems they will be able to connect relationships of equations in two ways. While solving these tasks in multiple ways I hope that they connect solving equations with the graphical intersection of those equations.

In the task students will be asked to solve a problem in more than one way. Solving in multiple ways is not something students are that comfortable doing as it is not commonly asked of them so directly. I would guess that most students will struggle to create the two initial equations for each problem. Beyond this, I believe many will struggle to solve the problem algebraically or graphically depending on which method they have become more comfortable.

The task could be modified for advanced students to have a linear equation and a quadratic function as the equations to show that you can have multiple solutions. For students who are struggling simpler equations, constant functions could be used or given as examples to extend upon to scaffold the students toward this concept.