Diego and Stefan 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, Diego has 20 signatures, and Stefan has 10. Diego is collecting signatures at an average rate of 3 per day, whereas Stefan is averaging 4 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?

Question

Write a system of equations, graph them, and type the solution.

Answer 1

To approach the problem, we'll define two variables to represent the number of days they collect signatures:

• Let \( d \) be the number of days until they have the same number of signatures.
• Let \( D \) be the total number of signatures Diego has after \( d \) days.
• Let \( S \) be the total number of signatures Stefan has after \( d \) days.

Step 1: Set up the equations

Given that:

• Diego starts with 20 signatures and collects 3 signatures per day, the equation for Diego's total signatures can be expressed as:

  \[ D = 20 + 3d \]

• Stefan starts with 10 signatures and collects 4 signatures per day, the equation for Stefan's total signatures is:

  \[ S = 10 + 4d \]

Step 2: Set the equations equal to each other

To find when they will have the same number of signatures, we set \( D \) equal to \( S \):

\[ 20 + 3d = 10 + 4d \]

Step 3: Solve for \( d \)

Rearranging the equation gives:

\[ 20 - 10 = 4d - 3d \]

\[ 10 = d \]

So, \( d = 10 \). This means it will take 10 days for Diego and Stefan to have the same number of signatures.

Step 4: Check the solution

Now let's calculate the number of signatures each will have after 10 days:

• For Diego: \[ D = 20 + 3(10) = 20 + 30 = 50 \]

• For Stefan: \[ S = 10 + 4(10) = 10 + 40 = 50 \]

Both Diego and Stefan will have 50 signatures after 10 days.

Conclusion

It will take 10 days for Diego and Stefan to have the same number of signatures.

Summary of the system of equations:

\[ \begin{align*} D & = 20 + 3d \ S & = 10 + 4d \ \end{align*} \] With the solution \( d = 10 \).