How Many Solutions?

5 min

Teacher Prep
Setup
Give students 2—3 minutes of quiet think time followed by a whole-class discussion. 

Narrative

Students extend their understanding from the previous lessons to recognize the structure of a linear equation for all possible types of solutions: one solution, no solution, or infinitely many solutions. Students are still using language such as “true for one value of xx,” “always true” or “true for any value of xx,” and “never true.” Students should be able to articulate that this depends both on the coefficient of the variable and on the constant term on each side of the equation.

Launch

Give students 2–3 minutes of quiet think time followed by a whole-class discussion. 

Student Task

Match each equation with the number of values that solve the equation.

  1. 12(x−3)+18=6(2x−3)12(x-3)+18=6(2x-3)
  2. 12(x−3)+18=4(3x−3)12(x-3)+18=4(3x-3)
  3. 12(x−3)+18=4(2x−3)12(x-3)+18=4(2x-3)
  • true for only 1 value
  • true for no values
  • true for any value

Sample Response

  1.  12(x−3)+18=6(2x−3)12(x-3)+18=6(2x-3) is true for any value.
  2.  12(x−3)+18=4(3x−3)12(x-3)+18=4(3x-3) is true for no values.
  3.  12(x−3)+18=4(2x−3)12(x-3)+18=4(2x-3) is true for only 1 value.
Activity Synthesis (Teacher Notes)

In order to highlight the structure of these equations, ask students:

  • “What do you notice about equations that are true for no values?” (These equations have equal or equivalent coefficients for the variable, but unequal values for the constants on each side of the equation.)
  • “What do you notice about equations that are true for all values?” (These equations have equivalent expressions on each side of the equation, so the coefficients are equal and the constants are equal or equivalent on each side.)
  • “What do you notice about equations that have exactly one value that makes them true?” (These equations have different values for the coefficients on each side of the equation and it doesn’t matter what the constant term says.)

Display the equation x=12x=12 for all to see. Ask students how this fits with their explanations. (We can see that there is one solution. Another way to think of this is that the coefficient of xx is 1 on the left side of the equation, and the coefficient of xx is 0 on the right side of the equation. So the coefficients of xx are different, just as the explanation states.)

Standards
Addressing
  • 8.EE.7.a·Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers).
  • 8.EE.C.7.a·Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form <span class="math">\(x = a\)</span>, <span class="math">\(a = a\)</span>, or <span class="math">\(a = b\)</span> results (where <span class="math">\(a\)</span> and <span class="math">\(b\)</span> are different numbers).

15 min

20 min