3.1 Systems of Linear Equations

Lesson Plan 3.1: Systems of Linear Equations

Objectives:

• Be able to recognize a system of linear equations.
• Be comfortable solving a system of linear equations using substitution, elimination, and graphing methods.

Suggested Lecture Breaks:

• MWF: You have three days. The first day, get through problem 3. The second day, at least start problem 5. Finish on the last day.
• MW/TR: You have two days. The first day, get through problem 4.

Suggested Lecture Notes:

• Define a system of linear equations:
  • A system of linear equations is a grouping of two or more linear equations.

• Ask the class what a solution to a system of linear equations in two variables is, both algebraically and geometrically. Include in this discussion how many solutions a system can have.

• Hopefully, the class will notice that this is not too different than what we did with solving linear equations in one variable (which could be thought of as a partially solved system of linear equations). Make this connection clear.

• Go over the two main methods used for solving systems: substitution and elimination. Work through the same example with each method, so that they can compare the two. Also, highlight that either method will always work when solving a system of two linear equations in two variables, so it is a matter of convenience when deciding which method to use (it might be nice to point out to them when one method is easier than the other).

• Tie in parallel and perpendicular lines as special cases of systems of two linear equations.

• Ask the class what a solution to a system of linear equations would be if the two lines in the system are parallel. Of course, there would be no solution in this case since the lines do not intersect. As a follow-up, ask whether this is the only time (graphically) that there is no solution.
• Next, ask them what a solution to a system of linear equations would be if the two lines are perpendicular. Point out that this is not the only way that a system has exactly one solution.

• Discuss what a solution to a system of linear equations would be if the two lines in the system are really the same line, i.e., the equations which define them are equivalent equations. Then, work through the following example so that students can see how WeBWorK wants them to enter their solutions in this case:

Suggested Example:

     \begin{align*}\left\{\begin{array}{rl}
        2x+4y&=8 
     	 4x+8y&=16   
     \end{array}\right.\end{align*}

Point out that after attempting to solve the system by elimination, we get the statement $0=0$ (or whatever tautology you arrive at), so we know that the two lines are really defined by equivalent expressions. WeBWorK wants them to write their solution as:

     \begin{align*}\left\{\begin{array}{rl}
       x&=x  
       y&= -\frac{1}{2}x+2
     \end{array}\right.\end{align*}

Explain why this form for an answer makes sense (we can let $x$ be anything we want, so $x=x$; we then can plug in whatever $x$ we choose into the $y$ equation and get an $(x,y)$-pair that satisfies both of our original equations).

Comments on the handouts:

• Question 5: It may be helpful to do a problem on the board exemplifying why you would choose one method over the other in solving. Be clear, however, that either method will give you the correct answer.