Let's work through each of the problems on the page one by one.

5. Marvin and Karuk study the equivalent equations $\frac{1}{3}x + 5 = 5$ and $x + 15 = 15$.

• Marvin's claim: He can form $x + 15 = 15$ by multiplying both sides of $\frac{1}{3}x + 5 = 5$ by 3.

  • Check: Multiply both sides of the equation $\frac{1}{3}x + 5 = 5$ by 3: $3 \times \left( \frac{1}{3}x + 5 \right) = 3 \times 5$ $x + 15 = 15$ Marvin's method is correct.

• Karuk's claim: He can form $x + 15 = 15$ by adding $\frac{2}{3}x$ and 10 to both sides of $\frac{1}{3}x + 5 = 5$.

  • Check: Adding $\frac{2}{3}x$ and 10 to both sides gives: $\frac{1}{3}x + 5 + \frac{2}{3}x + 10 = 5 + \frac{2}{3}x + 10$ $x + 15 = \frac{2}{3}x + 15$ This equation is incorrect, so Karuk is not correct.

Thus, Marvin is correct.

6. Describe the change made to each equation to make the statement true.

• (a) $4(x - 5) = 12$ is rewritten as $4x - 20 = 12$:

  • The change is distributing 4 to both terms inside the parentheses.

• (b) $\frac{5x - 4}{3} = 10$ is rewritten as $5x - 4 = 30$:

  • The change is multiplying both sides by 3.

• (c) $2x + 5 = x + 8$ is rewritten as $x + 5 = 8$:

  • The change is subtracting $x$ from both sides.

• (d) $3x + 8 = 2x - 7$ is rewritten as $3x + 15 = 2x$:

  • The change is adding 7 to both sides.

7. Match each equation on the left with an equivalent equation on the right.

• $3(x - 6) = 18 - 9x$ matches $3x - 6 = 9x + 6$. Simplify both sides: $3x - 18 = 18 - 9x \quad \text{(original)} \quad \text{and} \quad 3x - 6 = 9x + 6 \quad \text{(matched)}$ This is the equivalent equation.

• $x - 11 = -3 - 3x$ matches $x - 6 = 6 - 3x$.

• $x - 2 = 3x + 2$ matches $x - 2 = 6 - 3x$.

8. Chen says the equation $4\left(\frac{2}{3}x - 6\right) = 7 - 3x$ is equivalent to $\frac{8}{3}x - 24 = 7 - 3x$. Why is Chen not correct?

• To check this, distribute the 4: $4\left(\frac{2}{3}x - 6\right) = \frac{8}{3}x - 24$ So, Chen's multiplication is correct.

• However, this does not create an equivalent equation, because multiplying by 4 changes the balance of the equation. The original equation was $7 - 3x$, and this remains unchanged, so they are not equivalent.

Chen is not correct because his approach changes only one side of the equation.

Do you want more details on any of these? Here are 5 follow-up questions:

1. What happens if Karuk adds $\frac{2}{3}x$ only to one side in question 5?
2. Can you simplify $\frac{1}{3}x + 5 = 5$ in a different way?
3. Why does distributing change the form of an equation like in question 6(a)?
4. How can you verify the steps in question 7 with algebraic manipulation?
5. What property of equality would help correct Chen's approach in question 8?

Tip: Always ensure that operations are applied equally to both sides of an equation to maintain its balance.