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Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.

Yesterday, two friends went into a bank to open savings accounts. Maura started by putting $2 in her account, and she will deposit an additional $82 each week. Brad made no initial deposit, but he will add $83 more each week. In a few weeks, the friends will have the same account balance. What is that account balance? How many weeks will that take?

Maura and Brad will each have an account balance of $ ___ after ___ weeks.

5 answers

To model the situation, we can define the variables:

  • Let \( t \) represent the number of weeks.
  • Let \( M(t) \) be Maura's account balance after \( t \) weeks.
  • Let \( B(t) \) be Brad's account balance after \( t \) weeks.

From the information given:

  1. Maura's account balance after \( t \) weeks can be represented as: \[ M(t) = 2 + 82t \]

  2. Brad's account balance after \( t \) weeks can be represented as: \[ B(t) = 83t \]

We need to find \( t \) when Maura's account balance equals Brad's account balance, which gives us the equation: \[ M(t) = B(t) \]

Substituting the expressions we found: \[ 2 + 82t = 83t \]

Now, we will solve for \( t \):

  1. Rearranging the equation gives: \[ 2 = 83t - 82t \] \[ 2 = t \]

Now that we found \( t = 2 \) weeks, we can find the account balance of both Maura and Brad after 2 weeks.

Calculating Maura's account balance: \[ M(2) = 2 + 82(2) = 2 + 164 = 166 \]

Calculating Brad's account balance: \[ B(2) = 83(2) = 166 \]

Thus, Maura and Brad will each have an account balance of $166 after 2 weeks. Therefore, we fill in the blanks:

Maura and Brad will each have an account balance of $ 166 after 2 weeks.

Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.

Two kids at a summer camp, Isaiah and Linda, are competing in a potato sack race. Isaiah is younger, so he is given a head start of 26 yards. When the race starts, Isaiah hops at a rate of 1 yard per second, and Linda hops 3 yards per second. Eventually, Linda will overtake Isaiah. How long will that take? How far will Linda have to hop?

It will take
seconds for Linda to hop
yards and catch up to Isaiah.

To model the situation, we can define the variables:

  • Let \( t \) represent the time in seconds after the race starts.
  • Let \( I(t) \) be Isaiah's distance from the starting line after \( t \) seconds.
  • Let \( L(t) \) be Linda's distance from the starting line after \( t \) seconds.

From the information given:

  1. Isaiah has a head start of 26 yards, and he hops at a rate of 1 yard per second. His distance after \( t \) seconds can be represented as: \[ I(t) = 26 + 1t = 26 + t \]

  2. Linda hops at a rate of 3 yards per second with no head start. Her distance after \( t \) seconds can be represented as: \[ L(t) = 3t \]

We need to find \( t \) when Linda overtakes Isaiah, which gives us the equation: \[ I(t) = L(t) \]

Substituting the expressions we found: \[ 26 + t = 3t \]

Now, we will solve for \( t \):

  1. Rearranging the equation gives: \[ 26 = 3t - t \] \[ 26 = 2t \] \[ t = \frac{26}{2} = 13 \]

Now that we found \( t = 13 \) seconds, we can find out how far Linda hops in that time:

Calculating the distance Linda hops: \[ L(13) = 3(13) = 39 \text{ yards} \]

Thus, it will take 13 seconds for Linda to hop 39 yards and catch up to Isaiah. Therefore, we can fill in the blanks:

It will take 13 seconds for Linda to hop 39 yards and catch up to Isaiah.

Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.

Olivia wants to take group fitness classes at a nearby gym, but needs to start by selecting a membership plan. With the first membership plan, Olivia can pay $34 per month, plus $3 for each group class she attends. Alternately, she can get the second membership plan and pay $25 per month plus $6 per class. If Olivia attends a certain number of classes in a month, the two membership plans end up costing the same total amount. What is that total amount? How many classes per month is that?

Each membership plan costs $
if Olivia takes
classes per month.

To model the situation, we can define the following variables:

  • Let \( c \) represent the number of classes Olivia attends in a month.
  • Let \( A(c) \) be the total cost of the first membership plan.
  • Let \( B(c) \) be the total cost of the second membership plan.

From the information given, we can write the cost equations for each plan:

  1. For the first membership plan: \[ A(c) = 34 + 3c \]

  2. For the second membership plan: \[ B(c) = 25 + 6c \]

We are looking for the number of classes \( c \) where the two membership plans cost the same, which gives us the equation: \[ A(c) = B(c) \]

Substituting the expressions we found: \[ 34 + 3c = 25 + 6c \]

Now, we will solve for \( c \):

  1. Rearranging the equation gives: \[ 34 - 25 = 6c - 3c \] \[ 9 = 3c \]

  2. Dividing both sides by 3 gives: \[ c = 3 \]

Now that we found \( c = 3 \) classes, we can find the total amount for either membership plan:

Calculating the total cost for the first membership plan: \[ A(3) = 34 + 3(3) = 34 + 9 = 43 \]

Calculating the total cost for the second membership plan: \[ B(3) = 25 + 6(3) = 25 + 18 = 43 \]

Thus, each membership plan costs $43 if Olivia takes 3 classes per month. Therefore, we fill in the blanks:

Each membership plan costs $ 43 if Olivia takes 3 classes per month.