Let's break this down step by step.

Part 1: Write an equation that represents the number of people at the party.

The equation given is: $\frac{4}{5}x - 9 = 59$ where $x$ represents the total number of people at the party.

To solve for $x$, follow these steps:

1. Add 9 to both sides to isolate the term with $x$: $\frac{4}{5}x - 9 + 9 = 59 + 9$ $\frac{4}{5}x = 68$

2. Multiply both sides by the reciprocal of $\frac{4}{5}$, which is $\frac{5}{4}$: $x = 68 \times \frac{5}{4}$ $x = \frac{340}{4} = 85$

Thus, the number of people at the party is 85.

Part 2: Write a one-step equation that has the same solution.

We want to create a simple one-step equation that gives us the same result. If we start with the equation from Part 1 and simplify it, we can multiply both sides by $\frac{5}{4}$ to get: $\frac{4}{5}x = 68$ Multiply both sides by $\frac{5}{4}$: $x = 68 \times \frac{5}{4} = 85$

Thus, the one-step equation would be: $\frac{4}{5}x = 68$ which directly leads to the same solution.

Final Answer:

• The number of people at the party is 85.
• A one-step equation for the same solution is $\frac{4}{5}x = 68$.

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Would you like further explanation or help with any of the steps?

Related questions:

1. How do you solve equations with fractions like $\frac{4}{5}x - 9 = 59$?
2. What does "reciprocal" mean in algebraic equations?
3. How can you check the solution to an equation like $\frac{4}{5}x - 9 = 59$?
4. What other methods can you use to solve equations involving fractions?
5. How do you work with equations that involve both fractions and variables?

Tip: When working with equations that have fractions, always multiply both sides of the equation by the denominator to eliminate the fraction.