Let's analyze the given equations and determine if they are equivalent to the original equation, $17 = y - z$, by using the properties of equality.

Original equation:

$17 = y - z$

We will rewrite the equations and check if they match the original equation.

1. $90 = (y - z) \cdot 5$

We can divide both sides of the equation by 5: $90 \div 5 = (y - z) \quad \Rightarrow \quad 18 = y - z$

This is not equivalent to the original equation, which is $17 = y - z$. Thus, this equation is not correct.

2. $-95 = -5(y - z)$

We can divide both sides by -5: $-95 \div -5 = y - z \quad \Rightarrow \quad 19 = y - z$

This is also not equivalent to $17 = y - z$. Hence, this equation is not correct.

3. $-51 = -3(y - z)$

We divide both sides by -3: $-51 \div -3 = y - z \quad \Rightarrow \quad 17 = y - z$

This matches the original equation, so this equation is correct.

4. $34 = (y - z) \cdot 2$

We divide both sides by 2: $34 \div 2 = y - z \quad \Rightarrow \quad 17 = y - z$

This is also equivalent to the original equation, so this equation is correct.

Conclusion:

• The correct equations are:
  • $-51 = -3(y - z)$
  • $34 = (y - z) \cdot 2$

Would you like further details or have any additional questions?

Here are some related questions:

1. What are the properties of equality used in these examples?
2. How do you identify equivalent equations in general?
3. What happens if we multiply or divide both sides of an equation by zero?
4. Can equations with different constants on both sides be equivalent?
5. How does distributing constants affect equation equivalency?

Tip: Always perform the same operation on both sides of an equation to maintain its equivalence.