Unit 2, Lesson 7
Explaining Steps for Rewriting Equations
Algebra 1
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Is 0 a solution to each equation?
Here are some pairs of equations. While one partner listens, the other partner should:
Then switch roles until you run out of time or you run out of pairs of equations.
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Noah is having trouble solving two equations. In each case, he takes steps that he thinks are acceptable but ends up with statements that are clearly not true.
Analyze Noah’s work on each equation and the moves he made. Are they acceptable moves? Why do you think he ends up with a false equation?
Discuss your observations with your group and be prepared to share your conclusions. If you get stuck, consider solving each equation.
When solving an equation, sometimes we end up with a false equation instead of a solution. Let’s look at two examples.
Example 1:
Here are two attempts to solve it.
Each attempt shows acceptable moves, but the final equation is a false statement. Why is that?
When solving an equation, we usually start by assuming that there is at least one value that makes the equation true. The equation can be interpreted as: 4 groups of are equal to 4 groups of . There are no values of that can make this true.
For instance, if , then . It's not possible that 4 times 11 is equal to 4 times 10. Likewise, 1.5 is 1 more than 0.5, but 4 groups of 1.5 cannot be equal to 4 groups of 0.5.
Because of this, the moves made to solve the equation would not lead to a solution. The equation has no solutions.
Example 2:
Each step in the process seems acceptable, but the last equation is a false statement.
It is not easy to tell from the original equation whether it has a solution, but if we look at the equivalent equation , we can see that 0 could be a solution. When is 0, the equation is , which is a true statement. What is going on here?
The last move in the solving process was division by . Because 0 could be the value of and dividing by 0 gives an undefined number, we don't usually divide by the variable we're solving for. Doing this might make us miss a solution, namely .
Here are two ways to solve the equation once we get to :