(5) To find the value of $$ r $$, Stephen rewrites $$ 3 r-18=27 $$ as $$ 3 i r-61=319 $$. How could this help Stephen reason about the value of $$ r $$ ? Explain your reasoning. solution

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1. Stephen starts with the equation  
   $$
   3r - 18 = 27.
   $$
   To solve it normally one would add 18 to both sides and then divide by 3 to get  
   $$
   r = \frac{27 + 18}{3} = 15.
   $$

2. In rewriting the equation as  
   $$
   3\,r - 61 = 319,
   $$
   Stephen is using an equivalent transformation—that is, he is adding (or subtracting) the same numbers on both sides of the original equation (or performing a combination of operations that are equivalent to doing so).

3. The key idea is that any operation done equally on both sides does not change the solution. For example, if one adds the same number to both sides or multiplies (or divides) both sides by a nonzero number, the solution for $$ r $$ remains the same. Even though the numbers in the rewritten equation are different from the original, if the rewrite was done correctly using equivalent operations, the value of $$ r $$ will not change.

4. Thus, by comparing the two equations, Stephen can reason that although the constant terms have changed (from $$-18$$ and $$27$$ to $$-61$$ and $$319$$), the coefficient of $$ r $$ is still $$ 3 $$. This indicates that the operations he performed to rewrite the equation were applied equally to both sides. In other words, the transformation preserves the equality, and so the solution $$ r = 15 $$ for the original equation must also be the solution of the transformed equation.

5. Overall, rewriting the equation in a different form serves as a check on the reasoning: if after applying equivalent operations the form of the equation changes but the method for isolating $$ r $$ remains the same, Stephen can be confident that the value of $$ r $$ is unchanged and that his original solution is correct.
By rewriting the equation, Stephen can verify that the value of $$ r $$ remains the same, confirming that $$ r = 15 $$ is correct.

(5) To find the value of \( r \), Stephen rewrites \( 3 r-18=27 \) as \( 3 i r-61=319 \). How could this help Stephen reason about the value of \( r \) ? Explain your reasoning. solution

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