Solve general systems of linear equations in two variables
• Solve systems of linear equations exactly by algebraic methods (A-REI.C.6).
• Understand the principles behind the method of elimination (A-REI.C.5).
Students were introduced to the basic methods of solving systems of equations in middle school. In Section 3 of this unit they used simple systems to solve modeling problems. In this section they become fluent in general methods for solving systems algebraically and reason through the justification for these methods.
Tasks
1
Accurately weighing pennies I
WHAT: In 1982 the weight of the penny changed from 3.11 grams to 2.50 grams. Students are asked how many old and new pennies there are in a stack of 50 pennies weighting 138.42. Then they are asked similar questions given restrictions on the accuracy of the weight measurements for either the stack or the individual pennies.
WHY: Although the problem can be solved by trial and error, the number of decimal places in the coefficients motivates writing and solving a system of equations as a much more expeditious method. The system can be solved by substitution or by elimination, and there are strategic choices available for each method if the student observes the structure of the equations (MP7).
2
Estimating a Solution via Graphs
WHAT: A system of linear equations is given along with an incorrect student solution. Students give reasons why the solution must be incorrect, by reasoning quantitatively in terms of the slopes and intercepts, and by working with the graphs of the equations (MP2).
WHY: Part of becoming proficient with algebraic techniques for solving systems of linear equations is learning to detect errors. This task gives students an opportunity use quantitative and graphical reasons to detect an error in a solution (MP5). The equations have been chosen so that finding the exact solution requires significant calculation so that it is easy to make an error.
3
Solving Two Equations in Two Unknowns
WHAT: In this problem students analyze the elimination method used to solve a system of two equations with two unknowns. Students also look at the results of solving the same system through different steps noticing that the solution doesn’t change (MP8).
WHY: The goal of this task is to help students see the validity of the elimination method for solving systems of two equations in two unknowns. That is, the new system of equations produced by the method has the same solution(s) as the initial system. This is a subtle and vital point, though students should already be familiar with implementing this procedure before working on this task.
4
Accurately weighing pennies II
WHAT: Students are given the weights of three different pennies: those made between 1859 and 1864, between 1865 and 1982, and between 1983 and the present day. As in Accurately Weighing Pennies I students are given a roll of 50 pennies with a weight of 145 grams and must figure out how many of each type of penny is in the roll. The second and third parts look at the possibilities of two rolls having the same weight with a different make-up of pennies.
WHY: The problem leads to a system of two equations in three unknowns, an underdetermined system. Thus students must use additional information to solve the problem, namely that the number of coins must be an integer (MP1). The task is more appropriate for instructional use than assessment.