At the beginning of the year, Jose had \( \$ 30 \) in savings and saved an additional \( \$ 15 \) each week thereafter. Riley started the year with \( \$ 70 \) and saved \( \$ 10 \) every week. Let \( J \) represent the amount of money Jose has saved \( t \) weeks after the beginning of the year and let \( R \) represent the amount of money Riley has saved \( t \) weeks after the beginning of the year. Write an equation for each situation, in terms of \( t \), and determine the amount of money Jose and Riley have saved in the week that they have the same amount of money saved.

For Jose, the equation representing his savings over time is given as \( J(t) = 30 + 15t \), where \( t \) is the number of weeks since the beginning of the year. For Riley, her savings is represented by the equation \( R(t) = 70 + 10t \). To find out when they have the same amount saved, we set the two equations equal to each other: \[ 30 + 15t = 70 + 10t \] Solving for \( t \): 1. Subtract \( 10t \) from both sides: \[ 30 + 5t = 70 \] 2. Subtract \( 30 \) from both sides: \[ 5t = 40 \] 3. Divide by \( 5 \): \[ t = 8 \] Now, substituting \( t = 8 \) back into either equation to find their savings at that time: For Jose: \[ J(8) = 30 + 15(8) = 30 + 120 = 150 \] For Riley: \[ R(8) = 70 + 10(8) = 70 + 80 = 150 \] Thus, Jose and Riley both have \( \$150 \) saved after 8 weeks! Now, how about a little historical twist? Did you know that people have been saving money since ancient times? The earliest known banking system dates back to around 2000 BC in Mesopotamia, where temples and palaces would store grains and other valuables. People would deposit their goods and receive loans in return, laying the groundwork for the modern banking system we use today! And if you're curious about personal finance tips, remember that consistency is key! Setting up automatic transfers to your savings account can help you save without even thinking about it. Plus, track your expenses to identify areas where you can cut back and save more! Small changes can lead to big savings over time.