Writing Systems of Equations

10 min

Teacher Prep
Setup
Groups of 2. 2 minutes of partner work time followed by whole-class discussion.

Narrative

This Warm-up asks students to connect the algebraic representations of systems of equations to the number of solutions. Efficient students will recognize that this can be done without solving the system, but rather by using slope, yy-intercept, or other methods for recognizing the number of solutions.

Monitor for students who use these methods:

  1. Solve the systems to find the number of solutions.
  2. Use the slope and yy-intercept to determine the number of solutions.
  3. Manipulate the equations into another form, then compare the equations.
  4. Notice that the left side of the second equation in System C is double the left side of the first equation, but the right side is not.

Launch

Arrange students in groups of 2. Tell students that each number can be used more than once. Allow students 2 minutes of work time followed by a whole-class discussion.

Select work from students with different strategies, such as those described in the Activity Narrative, to share later.

Student Task

How many solutions does each system have? Be prepared to share your reasoning.

  1. {y=-43x+4y=-43x−1\begin{cases} y=\text-\frac43x+4 \\ y = \text-\frac43x-1 \end{cases}
  2. {y=4x−5y=-2x+7\begin{cases} y=4x-5 \\ y = \text-2x+7 \end{cases}
  3. {2x+3y=84x+6y=17\begin{cases} 2x+3y = 8 \\ 4x+6y = 17 \end{cases}
  4. {y=5x−15y=5(x−3)\begin{cases} y= 5x-15 \\ y= 5(x-3) \end{cases}

Sample Response

  1. 0
  2. 1
  3. 0
  4. infinite
Activity Synthesis (Teacher Notes)

The goal of this discussion is to compare strategies that students use to find the number of solutions for a system of equations. 

Invite a group of students to share their solution and reasoning. Then, ask:

  • “Did anyone solve the problem the same way, but would explain it differently?”
  • “How do the slope and intercept show up in each method?”
Standards
Addressing
  • 8.EE.8.b·Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. <em>For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6.</em>
  • 8.EE.C.8.b·Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. <span>For example, <span class="math">\(3x + 2y = 5\)</span> and <span class="math">\(3x + 2y = 6\)</span> have no solution because <span class="math">\(3x + 2y\)</span> cannot simultaneously be <span class="math">\(5\)</span> and <span class="math">\(6\)</span>.</span>

20 min