Question

Manuel and Ruben both have bank accounts. The table represents the balance, y, for each account after x weeks. Which system of equations represents the table? y = 11.5x and y = –13xy = 11.5x + 218 and y = –13x + 22y = 11.5x + 22 and y = –13x + 218y = 22x + 11.5 and y = 218x – 13 Number of Weeks Manuel’s Balance ($) Ruben’s Balance ($)
0 22 218
1 33.50 205
2 45 192
3 56.50 179

Answer 1

To determine which equations represent the table of balances for Manuel and Ruben after a certain number of weeks, we can analyze the data provided. The balances for each person after \( x \) weeks are as follows:

\[
\begin{array}{|c|c|c|}
\hline
\text{Number of Weeks} & \text{Manuel's Balance ($)} & \text{Ruben's Balance ($)} \\
\hline
0 & 22 & 218 \\
1 & 33.50 & 205 \\
2 & 45 & 192 \\
3 & 56.50 & 179 \\
\hline
\end{array}
\]

### Step 1: Determine the linear equation for Manuel’s balance.

We start with Manuel's balances:

- At \( x = 0 \), \( y = 22 \)
- At \( x = 1 \), \( y = 33.50 \)

To find the slope (\( m \)):
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{33.50 - 22}{1 - 0} = 11.50
\]

Using the point-slope form of the equation:
\[
y - y_1 = m(x - x_1)
\]
Using the point \( (0, 22) \):
\[
y - 22 = 11.5(x - 0) \implies y = 11.5x + 22
\]

### Step 2: Determine the linear equation for Ruben’s balance.

Now, for Ruben's balances:

- At \( x = 0 \), \( y = 218 \)
- At \( x = 1 \), \( y = 205 \)

Again, we calculate the slope:
\[
m = \frac{205 - 218}{1 - 0} = \frac{-13}{1} = -13
\]

Using the point-slope form:
\[
y - y_1 = m(x - x_1)
\]
Using the point \( (0, 218) \):
\[
y - 218 = -13(x - 0) \implies y = -13x + 218
\]

### Step 3: Write the system of equations.

Now we combine the two equations derived:
1. Manuel’s balance: \( y = 11.5x + 22 \)
2. Ruben’s balance: \( y = -13x + 218 \)

This means the system of equations is:
\[
\begin{cases}
y = 11.5x + 22 \\
y = -13x + 218
\end{cases}
\]

### Conclusion

The correct system of equations that represents the balances for Manuel and Ruben is:

\[
\boxed{y = 11.5x + 22 \text{ and } y = -13x + 218}
\]

Answer 2

Manuel and Ruben both have bank accounts. The system of equations models their balances after x weeks.

y = 11.5x + 22

y = –13x + 218

Their balances will be the same after___
weeks.

Their balances will be $___
.

Answer 3

To find out when Manuel's and Ruben's balances will be the same, we need to set their equations equal to each other and solve for \( x \).

Given the equations:

1. Manuel's balance: \( y = 11.5x + 22 \)
2. Ruben's balance: \( y = -13x + 218 \)

### Step 1: Set the equations equal to each other

\[
11.5x + 22 = -13x + 218
\]

### Step 2: Solve for \( x \)

First, add \( 13x \) to both sides:

\[
11.5x + 13x + 22 = 218
\]

Combine like terms:

\[
24.5x + 22 = 218
\]

Now, subtract 22 from both sides:

\[
24.5x = 218 - 22
\]
\[
24.5x = 196
\]

Now, divide both sides by 24.5:

\[
x = \frac{196}{24.5} \approx 8
\]

So, their balances will be the same after approximately \( 8 \) weeks.

### Step 3: Find the balance at that time

We can substitute \( x = 8 \) back into either equation to find the balance. Using Manuel's equation:

\[
y = 11.5(8) + 22
\]

Calculating this gives:

\[
y = 92 + 22 = 114
\]

### Conclusion

Thus, Manuel's and Ruben's balances will be the same after **8 weeks**, and their balances will be **$114**.

So, the final answers are:

- Their balances will be the same after **8** weeks.
- Their balances will be **$114**.