Some students may say that doesn’t describe the relationship between the ages because it doesn’t involve the number 5. Encourage students to think of some values for and that do make the equation true. After they find some pairs of and , ask them what they notice about the pairs.

Select previously identified students to share their explanation of why Tyler's claim is true, in the order shown in the Activity Narrative. If no students mention the last approach, bring it up.

Next, emphasize two main points:

• Equivalent equations have exactly the same solutions, or exactly the same values that make each of the equations true. All of the equations in the last question are equivalent because they have 3 as the solution and they all have no other solutions.
• Suppose we start with a true equation, where the two sides are equal. If we perform the same operation to both sides of the equation and get a new equation where the two sides are also equal, we can say that the two equations are equivalent. For instance:
  • Subtracting both sides of by 6 gives . If and 15 are equal, then the expressions or numbers we get by subtracting 6 from each one are also equal. We can conclude that and are equivalent.
  • Dividing both sides of by 3 gives . If is equal to 15, the result of dividing by 3 is equal to dividing 15 by 3. We can conclude that and are equivalent.

Ask students:

• "How can we show that is equivalent to ?" (Multiplying both sides of by 9 gives .)
• "How can we show that is equivalent to ?" (Subtracting 2 from both sides of and then multiplying both sides of the resulting equation, , by gives .)

If no students notice that we have made these moves when solving equations, bring it to their attention. Highlight that solving an equation essentially involves writing a series of equivalent equations that eventually isolates the variable on one side.

Display the original equation and Equations A–F for all to see. Invite students to share what was done to the original equation to get each of those equations and whether they have the same solution. Along the way, compile a list of moves that lead to equations with the same solution and those that lead to different solutions. 

Ask students to observe the list and see what kinds of moves produce the Equations A–C and D–F. Record the moves that create equations with the same solutions, such as:

• Adding or multiplying both sides of the equal sign by the same number
• Applying properties of operations (commutative, associative, or distributive)
• Combining like terms

Also record the moves that create equations with different solutions, such as:

• Adding or multiplying a different number to the two sides, or performing an operation to only one side
• Adding different expressions to the two sides
• Performing different operations on each side

(The lists don't need to be comprehensive because students will examine these moves more closely later.)

Next, give students a couple of examples of how the equations and their solutions could be interpreted in context. For example:

• Equation A shows the cost calculation in cents, so the original price of a pair of jeans is unchanged.
• Equation D shows the discounted price, excluding tax, to be \$56.70, but in the original equation, the price \$56.70 including tax. This means that, in Equation D, the original price of one pair of jeans is different than in the first equation. 

Prompt students to interpret 1–2 other equations and to explain why the solution in each equation (or the price of one pair of jeans) is equal or unequal to that in the initial equation. Here are possible interpretations:

• Equation B shows the discounted price before sales tax to be \$54, which is \$2.70 less than \$56.70. This is also the case in the initial equation, so the original price of the jeans is still the same. 
• Equation C shows the discounted price as 90% of the original price, which is the same as 10% less than the original price (100%). The original price is unchanged.
• Equation E shows the discounted price, before tax, to be \$59.40. This is more than the pre-tax price in the initial equation, so the original price of the jeans here must be higher.
• Equation F shows the price of 2 pairs of jeans, including the discount and tax, to be \$56.70. This means the price of a pair of jeans must be much less than in the initial equation.