Match each equation with the number of values that solve the equation.
true for only 1 value
true for no values
true for any value
11.2
Activity
Your teacher will give you a set of cards containing equations.
Sort the cards into categories of your choosing.
Describe the defining characteristics of the categories, and be prepared to share your reasoning with the class.
11.3
Activity
For each equation, determine whether it has no solutions, exactly one solution, or is true for all values of (and has infinitely many solutions). If an equation has one solution, solve the equation to find the value of that makes the statement true.
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What do you notice about equations with one solution? How is this different from equations with no solutions and equations that are true for every ?
Student Lesson Summary
Sometimes it's possible to look at the structure of an equation and tell if it has infinitely many solutions or no solutions. For example, look at
Using the distributive property on the left and right sides, we get
From here, collecting like terms gives us
Without doing any more moves, we know that this equation is true for any value of because the left and right sides of the equation are the same.
Similarly, we can sometimes use structure to tell if an equation has no solutions. For example, look at
If we think about each move as we go, we can stop when we realize there is no solution:
Because the coefficient of is 6 on each side, we know that there is either no solution or infinitely many solutions. The last move makes it clear that the constant terms on each side, 5 and , are not the same. Because adding 5 to an amount is always less than adding to that same amount, we know that there are no solutions.
Doing moves to keep an equation balanced is a powerful part of solving equations, but thinking about what the structure of an equation tells us about the solutions is just as important.
In an expression like , the number 2 is called the constant term. It doesn’t change when the variable changes.
