Chau currently has $$ \$ 42 $$ in savings. He had been saving $$ \$ 7 $$ each week. Yesterday he spent $$ \$ 77 $$ of the savings. For how many weeks had he beer (a) Write an equation that could be used to answer the question above. First, choose the appropriate form. Then, fill in the blanks with the numbers 42,7 , and 77 . Let $$ w $$ represent the number of weeks. $$ \square w+\square=\square $$ (b) Solve the equation in part (a) to find the number of weeks. w $$ \quad \square $$

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UpStudy AI · Accepted Answer

Let's break down the problem step by step.

### Part (a): Write the Equation

1. **Identify the known values:**
   - Initial savings: $$ 42 $$
   - Amount saved each week: $$ 7 $$
   - Amount spent: $$ 77 $$

2. **Understanding the situation:**
   - Chau's total savings after saving for $$ w $$ weeks would be $$ 42 + 7w $$.
   - After spending $$ 77 $$, his remaining savings would be $$ 42 + 7w - 77 $$.

3. **Setting up the equation:**
   - We want to find out when his savings after spending $$ 77 $$ is equal to zero (or when he has no savings left).
   - Therefore, the equation can be set up as:
   $$
   7w + 42 - 77 = 0
   $$

4. **Filling in the blanks:**
   - The equation in the required form is:
   $$
   7w + 42 - 77 = 0
   $$
   Which can be simplified to:
   $$
   7w - 35 = 0
   $$

So, the equation is:
$$
\boxed{7w + 42 - 77 = 0}
$$

### Part (b): Solve the Equation

Now, let's solve the equation $$ 7w - 35 = 0 $$.

1. **Rearranging the equation:**
   $$
   7w = 35
   $$

2. **Dividing both sides by 7:**
   $$
   w = \frac{35}{7}
   $$

3. **Calculating the value:**
   $$
   w = 5
   $$

Thus, the number of weeks Chau had been saving is:
$$
\boxed{5}
$$
The equation is $$ 7w + 42 - 77 = 0 $$, and the number of weeks is 5.

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