Let's break each problem down by first forming the equation and then solving it.

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(a) Three more than twice a number is seven more than the number.

1. Let the number be $x$.
2. "Twice a number" is $2x$.
3. "Three more than twice a number" is $2x + 3$.
4. "Seven more than the number" is $x + 7$.

The equation based on the problem is: $2x + 3 = x + 7$

Now, solve for $x$: $2x + 3 = x + 7$ Subtract $x$ from both sides: $x + 3 = 7$ Subtract 3 from both sides: $x = 4$

So, the number is $\boxed{4}$.

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(b) Eight more than three times a number is four more than the number.

1. Let the number be $x$.
2. "Three times a number" is $3x$.
3. "Eight more than three times a number" is $3x + 8$.
4. "Four more than the number" is $x + 4$.

The equation is: $3x + 8 = x + 4$

Now, solve for $x$: $3x + 8 = x + 4$ Subtract $x$ from both sides: $2x + 8 = 4$ Subtract 8 from both sides: $2x = -4$ Divide both sides by 2: $x = -2$

So, the number is $\boxed{-2}$.

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(c) Six more than twice a number is four more than four times the number.

1. Let the number be $x$.
2. "Twice a number" is $2x$.
3. "Six more than twice a number" is $2x + 6$.
4. "Four times the number" is $4x$.
5. "Four more than four times the number" is $4x + 4$.

The equation is: $2x + 6 = 4x + 4$

Now, solve for $x$: $2x + 6 = 4x + 4$ Subtract $2x$ from both sides: $6 = 2x + 4$ Subtract 4 from both sides: $2 = 2x$ Divide both sides by 2: $x = 1$

So, the number is $\boxed{1}$.

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(d) Nine less than five times a number is three less than twice the number.

1. Let the number be $x$.
2. "Five times a number" is $5x$.
3. "Nine less than five times a number" is $5x - 9$.
4. "Twice the number" is $2x$.
5. "Three less than twice the number" is $2x - 3$.

The equation is: $5x - 9 = 2x - 3$

Now, solve for $x$: $5x - 9 = 2x - 3$ Subtract $2x$ from both sides: $3x - 9 = -3$ Add 9 to both sides: $3x = 6$ Divide both sides by 3: $x = 2$

So, the number is $\boxed{2}$.

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Summary of Answers:

• (a) $x = 4$
• (b) $x = -2$
• (c) $x = 1$
• (d) $x = 2$

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Would you like further explanations or details for any of these steps? Let me know!

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Here are some related questions you might find helpful:

1. What happens if we modify the constant terms in these equations?
2. How can we check our solutions algebraically once we've solved for $x$?
3. How would you approach solving equations with more variables?
4. What are some different ways of expressing these types of word problems mathematically?
5. What role does isolating the variable play in solving equations?

Tip: Always double-check the wording of the problem to make sure you're translating the conditions accurately into the equation!