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

Question

Florence and Mackenzie each want to run for president of their school's student body council. In order to do so, they must collect a certain number of signatures and get a nomination. So far, Florence has 24 signatures, and Mackenzie has 20. Florence is collecting signatures at an average rate of 5 per day, whereas Mackenzie is averaging 7 signatures every day. Assuming that their rate of collection stays the same, eventually the two will have collected the same number of signatures. How long will that take? How many signatures will they both have?

In
days, Florence and Mackenzie will each have collected
signatures.

Answer 1

To model this situation, we can create a system of equations based on the information provided.

Let \( x \) be the number of days they will continue to collect signatures.

1. The total number of signatures Florence will have after \( x \) days can be expressed as: \[ F(x) = 24 + 5x \] where 24 is the number of signatures she currently has, and 5 is the rate at which she collects signatures each day.

2. The total number of signatures Mackenzie will have after \( x \) days can be expressed as: \[ M(x) = 20 + 7x \] where 20 is the number of signatures she currently has, and 7 is her daily collection rate.

To find out when they will have the same number of signatures, we set the two equations equal to each other: \[ 24 + 5x = 20 + 7x \]

Now, we solve for \( x \):

1. Subtract \( 5x \) from both sides: \[ 24 = 20 + 2x \]

2. Subtract 20 from both sides: \[ 4 = 2x \]

3. Divide both sides by 2: \[ x = 2 \]

Now that we know it will take 2 days, we can calculate how many signatures they will both have at that time.

Substituting \( x = 2 \) back into either signature equation:

For Florence: \[ F(2) = 24 + 5(2) = 24 + 10 = 34 \]

For Mackenzie: \[ M(2) = 20 + 7(2) = 20 + 14 = 34 \]

Both Florence and Mackenzie will have collected 34 signatures after 2 days.

Thus, the final answer is:

In 2 days, Florence and Mackenzie will each have collected 34 signatures.